Tolkien never uses the term "age(s) of the sun" in his writings
Difficult to prove a negative, but at least he doesn't use it anything that has been published, and easily searchable. Certainly nothing that predates that fan usage of the terms.
Tolkien's "Ages" when given categories, are usually "Ages of the Children of Ilúvatar". (Though sometimes also "of Time" or "of the World") As such, the first age begins with the awakening of the elves.
In those days, in the Year one thousand and fifty of the Valar, the Elves awoke in Kuiviénen and the First Age of the Children of Ilúvatar began.
Morgoth's Ring - 'The Annals of Aman' - §10
In the manuscript as it was originally written the Elder Days began with the Awakening of the Elves: 'Here begin the Elder Days, or the First Age of the Children of Iluvatar'
The War of the Jewels - Christopher describing 'The Tale of Years'
The First Age begins with the Awaking and ends with the Downfall of Angband.
The Nature of Middle-earth - 'Generational Schemes'
The chronologies that Tolkien writes which begin the years from the rising of the sun use unit labels like "Years of the Sun", and not "First Age". The chronologies which do use unit labels of First Age years all have the Awakening of the Elves in year one.
First Age 1 - Quendi awake in the Spring (144 in number). Melian warned in a dream leaves Valinor and goes to Endor.
The Nature of Middle-earth - 'Key Dates'
Days of Bliss. End of Waiting Time. First Age begins. Awaking of Quendi (Spring).
1 - Awaking of Quendi (Spring).
The Nature of Middle-earth - 'Key Dates'
Tolkien also was concerned with having all three ages be of comparable length, rather than the first age just being a few centuries.
The Trees flower for 864 VY before Awaking = 124,416 [sun-]years. Quendi then awake in Spring of [VY] 865 (124,417 [YS]). “DB” still goes on, but Quendi start reckoning of First Age with Awaking.
First Age must last somewhat longer than SA (= 3,441). Still be more regularly “duodecimal” (as mythological) up to Death of Trees and after! Say, 4,056 years.
The Nature of Middle-earth - 'Key Dates'
The events contemporary with the rising of the Sun have also been described by both Tolkien and Christopher as the "end" of the first age, not the beginning.
[Quenya] was no longer a birth-tongue, but had become, as it were, an ‘Elvenlatin’, still used for ceremony, and for high matters of lore and song, by the High Elves, who had returned in exile to Middle-earth at the end of the First Age.
The Lord of the Rings Appendix F - Of the Elves
The title of this second part, The War of the Jewels, is an expression that my father often used of the last six centuries of the First Age: the history of Beleriand after the return of Morgoth to Middle-earth and the coming of the Noldor, until its end.
The War of the Jewels - Introduction
So where does the term "Ages of the Sun" come from?
There are a couple of places that may be to blame here.
For one, Tolkien has been a lot less clear in his published works about where the first age started than where it ended. It can be inferred, but it's not obvious.
For another, there were some early chronology resources, such as the one included in Robert Foster's 1978 "The Complete Guide to Middle-earth", which began with the rising of the Sun. Foster himself never uses the term "ages of the sun", and is very clear to say that his chronology was only a part of the first age, the part that was easily quantifiable given the materials available at the time. But perhaps this could have contributed a bit to the confusion.
However, others of course were a lot less careful than Foster was. And I think it is fair to say that the blame is not at all on Foster or Tolkien here.
As best as I can tell, the first person, and also the most notable person, to use this term was David Day, in his 1979 book A Tolkien Bestiary.
Day uses the term "ages of the sun" a few times in the body of this book. (Minimizing these quotations as much as possible to avoid spreading any more of Day's misinformation than needed).
the second [chart] is a closer view of the Ages of Sun ... for a detailed listing of events and incidents of the Second and Third Ages of the Sun, the reader must consult ...
A Tolkien Bestiary - Intro
In the first Age of the Sun the Werewolf race was bred.
A Tolkien Bestiary - Wolves
Day also includes a chart in the book, full of made up terms for different time periods.
A Tolkien Bestiary - A Chronology of Middle-earth and the Undying Lands. (PLEASE DO NOT RELY ON THIS OR ANYTHING ELSE FROM DAVID DAY)
David Day of course is never satisfied with simply making up information once, but likes repeating this over and over again. His countless other books continue to use these terms over and over again. I singled out A Tolkien Bestiary because it seems to be the first published place. I will not be going through all his other books here.