NOAA Solar Calculator
To find the positions of the sun in the sky for any given date and time the common method is to use the NOAA Solar Calculator. It is often alleged that the NOAA Solar Calculator is based on Round Earth Theory and Heliocentricism, and that because it can predict some attributes of the sun, it therefore means that the Round Earth Theory must true.
Firstly, these claims are universally made without demonstration that the calculator is entirely correct in all instances. A source for records of observations, comprehensive or not, is, of course, never given.
Secondly, the assumption that the calculator is based on Heliocentrism is seen to fall flat on reading of the details. The calculations are simple equations that are based on the pattern of previous observations and occurrences. By studying the repeating pattern of behavior of astronomical bodies over a period of time it is possible to create an equation that will predict when the next occurrence will occur or how it will behave in the future.
This is how Aristotle predicted astronomical occurrences, this is how the flat earth civilization known as the Ancient Babylonians predicted astronomical occurrences, and this is how "Modern" Astronomers predict astronomical occurrences today.
Contents
 1 Online Calculator Link
 2 Excel Worksheet Version
 3 Astronomical Algorithms
 3.1 Prediction Worked from the Bodies as They Are Seen in the Sky
 3.1.1 Geocentric: as seen from the center of the Earth
 3.1.2 Chapter: Rectangular Coordinates of the Sun
 3.1.3 Chapter: Equinoxes and Solstices
 3.1.4 Chapter: Equation of Time
 3.1.5 Chapter: Ephemeris for Physical Observations of the Sun
 3.1.6 Chapter: Position of the Planets
 3.1.7 Chapter: Elliptic Motion
 3.1.8 Chapter: Nearparabolic Motion
 3.1.9 Chapter: The Calculation of some Planetary Phenomena
 3.1 Prediction Worked from the Bodies as They Are Seen in the Sky
 4 Sun Triangulation Problem
 5 See also:
Online Calculator Link
Online Calculator Link: https://www.esrl.noaa.gov/gmd/grad/solcalc/
Excel Worksheet Version
Under the "Calculation Details" link the NOAA has provided an Excel spreadsheet version of their online calculator here:
https://www.esrl.noaa.gov/gmd/grad/solcalc/NOAA_Solar_Calculations_day.xls (Archive)
Feel free to look at the formula sources in the spreadsheet above and try and find where the Round Earth Theory is expressed or where we see Keplerian orbital mechanics. The calculations are simple equations that are based on the ancient method of assessing the pattern of astronomical occurrences to predict future behavior.
Heliocentric Elements
AU Column
There does appear to be a seemingly Heliocentric element in the worksheet.
See Column O with the title "Sun Rad Vector (AUs)"
The worksheet default is 1.000001018.
AU is short for 'Astronomical Unit', the distance between the sun and the earth in the Heliocentric Round Earth Theory, and Rad is short for 'Radius'.
Put 0 into that column and see what happens. It doesn't affect the predictions at all. Also try 9.5 AUs. No effect. The same results are seen result whether the calculator is operating under the assumption of 0 Astronomical Units or 9.5 Astronomical Units. Looking closer at the equations for this field, the AU field appears to be an output variable, not an input variable. The data of the sun's position in the sky and the simple trending seen in the formulas is not being created from this element. The data is deriving slight adjustments to the default 1.000001018.
AU Zero Out Example
To illustrate the above, lets first see the default data for the first default data row, 06:00, on the default date of 6/21/2010.
06:00 on 6/21/2010 Sun Rad Vector (AUs) 1.016240085 Solar Zenith Angle (deg) 115.2457185 Solar Elevation Angle (deg) 25.24571849 Solar Azimuth Angle (deg cw from N) 345.8691023
Next we zero out the AU field and see that the attributes do not change.
06:00 on 6/21/2010 Sun Rad Vector (AUs) 0 Solar Zenith Angle (deg) 115.2457185 Solar Elevation Angle (deg) 25.24571849 Solar Azimuth Angle (deg cw from N) 345.8691023
We then undo our changes and change only the month from 06/21 to 10/21.
06:00 on 10/21/2010 Sun Rad Vector (AUs) 1.016240085 Solar Zenith Angle (deg) 149.4436002 Solar Elevation Angle (deg) 59.44360016 Solar Azimuth Angle (deg cw from N) 341.0653868
Other Elements
Other elements such as Right Ascension, Declination, Azimuth, and Altitude are merely terms coined by astronomers to describe where things are in the sky.
Astronomical Algorithms
The online NOAA Solar Calculator and the associated Excel worksheet version are based on equations from a book:

A PDF of the above book is found here.
The following is found in the glossary:
This is interesting. The terminology used in the book suggests to the reader that the book is based on the heliocentric theory. However, as we can even delete the AU column entirely from the previously mentioned NOAA Solar Calculator Excel worksheet, and see that the worksheet still gives the same results for the sun, even when the year and day is changed to a future date after the column is removed, shows us that the element is useless in prediction of the sun's location.
The orbital mechanical calculations of objects with "mass," "orbiting" the sun, and dependent on a "gravitational constant," like described in the above definition and elsewhere in the book is not reflected in the simple equations in the Excel worksheet.
In some areas of the book the author appears to enjoy pretending that he is using Heliocentric Theory for his algorithms, and liberally uses heliocentric terms and factoids to give the impression that Astronomical Algorithms is an a masterpiece of Astronomy prediction. Yet, in truth, when the work is read closely the algorithms are revealed to be just statistical (patternbased) methods:
Prediction Worked from the Bodies as They Are Seen in the Sky
As the Ancients predicted astronomical events based on patterns, the solutions given in the book for predicting astronomical events are based on pattern trending and statistical analysis.
Throughout the work you will see that various problems are worked (patterns created from) geocentrically, rather than starting from the Heliocentric system to make a prediction. The definition for "geocentric" is given on p. 263:
Geocentric: as seen from the center of the Earth
In Flat Earth equivalent for this is, of course, the apparent arc that bodies are seen to make as they pass over the observer.
The problems in the book are not worked out using Orbital Mechanics or anything inherent in a Heliocentric System. The words "heliocentric" are used in various forms in the work, but upon close analysis, heliocentric is meant as the assumed heliocentric attributes the bodies are imagined to have when viewed from the earth. Some of the observations are also used to calculate various properties of the assumed heliocentric model (the AU of Venus as described in the Positions of the Planets section, for example), but this is not the same as using the raw Heliocentric model to predict the positions of bodies. The elements calculated are used as output variables.
Another way heliocentric elements are used is to give the reader background information on how things are supposed to be.
From the Introduction on Page 2 the author admits as much

Several example below highlight this:
Chapter: Rectangular Coordinates of the Sun
From the first page of the chapter on p. 159
"The rectangular geocentric coordinates X,Y,Z of the Sun are needed for the calculation of an ephemeris of a minor planet or a comet. The origin of these coordinates is the center of the Earth."
As we can see, the coordinates of the sun are computed geocentricallyas in how it appears in the sky from the earth
Chapter: Equinoxes and Solstices
"The times of the equinoxes and solstices are the instants when the apparent geocentric longitude of the Sun (that is, calculated by including the effects of aberration and nutation) is a multiple of 90 degrees. (Because the latitude of the Sun is not exactly zero, the declination of the sun is not exactly zero at the instant of an equinox.)"
Again, we see that the method is worked out based on the view of the sun in the sky.
Chapter: Equation of Time
Page 171
The particular chapter starts off talking about a heliocentric model:
Due to eccentricity of its orbit, and to much less degree due to perturbations of the Moon and the planets, the Earth's heliocentric longitude for not vary uniformly. It follows that the Sun appears to describe the ecliptic at a nonuniform rate. Due to this, and also to the fact that the Sun is moving in the ecliptic and not along the celestial equator, its right ascension does not increase uniformly.
The chapter then switches to a view from the earth:
Consider a first fictitious Sun traveling along the ecliptic with a constant speed and coinciding with the true Sun at the perigee and apogee (the Earth is in perihelion and aphelion, respectively). Then consider a second fictitious Sun traveling along the celestial equator at a constant speed and coinciding with the first fictitious Sun at the equinoxes. This second fictitious Sun is the mean Sun, and by definition its right ascension increases at a uniform rate. (That is, there are no periodic terms, but its expression contains small secular terms in T2, T3, ...) When the mean Sun crosses the observer's meridian, it is mean noon there. True noon is the instant when the true Sun crosses the meridian. The equation of time is the difference between the hour angles of the true Sun and the mean Sun. Defined in this manner, the equation of time E, at a given instant, is given by... <various equations follow>
Note that the author gradually switches into a firstperson viewpoint, first by saying that the Sun travels along the celestial equator, which is seen as a line that passes across the sky, and then at a point inserts "the observer"...
The celestial equator is, of course, a line that passes across the sky as seen by the observer.
Chapter: Ephemeris for Physical Observations of the Sun
Page 177
The formulae given in this Chapter are based on the elements determined by Carrington (1863), which have been in used for many years. For a given instant, the requited quantities are: P = the position angle of the northern extremity of the axis of rotation, measured eastwards from the North Point of the solar disk ; B0 = the heliographic latitude of the center of the solar dist ; L0 = the heliographic longitude of the same point.
https://www.merriamwebster.com/dictionary/heliographic
Definition of heliographic from MarriamWebster
heliographic adjective he·lio·graph·ic \ ?hele?'grafik \
 measured on the sun's disk
Chapter: Position of the Planets
For the position of the planets we are led in with the idea that we are going to see a method for calculating the heliocentric coordinates of the planets:
"in 197 Bretagnon and Francous constructed the version called VSOP87, which gives periodic terms for calculating the planets' heliocentric coordinates directly, namely L, the ecliptical longitude B, the ecliptical latitude R, the radius vector (= distance to the Sun) It should be noted that L is really the planet's ecliptical longitude, not the orbital longitude.
From https://www.britannica.com/science/astronomicalmap#ref510162:

The chapter continues:
To obtain the heliocentric ecliptical longitude L of a planet at a given instant, referred to the mean equinox of the date, proceed as follows, Calculate the sum L0 of the terms of the series L0, the sum of L1 of the terms of series L1, then the longitude given in radians is given by L = (L0 + L1 T + L2 T^2 + L3 T^3 + L4 T^4 + L5 T^5)/10^8 Proceed in the same way for the heliocentric latitude B and for the radius vector R.
Notice in the above that the equation that we are calculating B and R, not using it to predict anything.
Continuing on in the chapter:
We calculate the heliocentric latitude B and the radius vector R in the same way. It should be noted that, in the case of Venus the series B5 and R5 do not exist. The results are B = 0.0457399 radian = 2o.62070, R = 0.724603 AU"
R, the distance of the object to the sun, is only used as an output variable to find where Venus would be in the Heliocentric Theory. It is not an input variable. The method, revealed for what it is, does not use the Heliocentric Theory to predict anything. It merely uses the observations of bodies as seen from earth to predict what the AU of Venus would be in the Heliocentric model.
Chapter: Elliptic Motion
Page 209
"In this Chapter we will describe two methods for the calculation of geocentric portions in the case of an elliptic orbit"
Chapter: Nearparabolic Motion
"The calculation of the geocentric places can then be performed as for the elliptic and parabolic motions."
Chapter: The Calculation of some Planetary Phenomena
Page 237
Greatest elongation of Mercury and Venus The value of the greatest elongation from the Sun is expressed in degrees and decimals. It concerns the maximum angular distance from the planet to the center of the Sun's disk, not the greatest difference between the geocentric ecliptical longitudes of the two bodies. There is no 'official' definition for the elongation of a planet to the Sun, and two different definitions could be considered: (a) the angular distance between the object and the center of the solar disk; (b) the difference between the geocentric longitudes of the object and the center of the solar disk Both definitions are used in astronomical literature. Definition (a) has been used in Astronomical Ephmeris since its beginning in 1960, and from 1981 onwards in its successor, the Astronomical Almanac. It is this definition we prefer. For example, for the visibility of Venus near its inferior conjunction, the important factor is not the longitude difference with the Sun, but the angular separation.
Sun Triangulation Problem
The sun positions given by the NOAA Solar Calculator cannot be used to determine the position of the sun in the Heliocentric system.
Video: Calculate the distance to the Sun (Where is the Sun?)
https://www.youtube.com/watch?v=9puRZH0i6Sc
Runtime: 15 Min
From the description (minor phrasing edits):

The author followed up with a couple of additional videos in 2018:
Video: Where is the Sun 2018 Update #1
https://www.youtube.com/watch?v=6H0qyvu9XQ
Runtime: 40 Min
Video: Where is the Sun 2018 Update #2
https://www.youtube.com/watch?v=dt3rxtVmWJc
Runtime: 17 Min