"First Age of the Sun" misconception

[Ar-Zigûr]

Why are so many people mistakenly identify the beginning of the First Age with the first sunrise? For example, there're this remark of Christopher Tolkien in his foreword to The War of the Jewels:

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.​

Even in the distant 1978, before the publication of the HoME's volumes, Rober Foster never suggested that the First Age began with the first sunrise. This is the entry "First Age" from his Complete Guide to Middle-earth:

FIRST AGE The first of the recorded ages of Middle-earth. The First Age may have begun with the completion of Arda or with the awakening of the Elves. Its early history, save for the Battle of the Powers and the Great Journey, is obscure, for most of the Eldar were in the West, and other races kept few records. Moreover, until the rising of the Moon and Sun there seems to have been no count of years in Middle-earth.
This was the Age of the Eldar and of Morgoth. It ended with the Wars of Beleriand and the Great Battle, in which Morgoth was cast out of Eä.
Also called the Elder Days and the Eldest Days.

This is an appendix with a chronology of the First Age in full from the same book:

In preparing this Chronology (for reasons explained in the Introduction) I have encountered a number of difficulties. First, Quenta Silmarillion provides no quantitative indication of the passage of time before the creation of the Sun and the return of the Noldorin Exiles; indeed, before the creation of the Two Trees the Valar apparently did not concern themselves with the measurement of Time. Therefore, the seventeen sections of the Chronology marked by Roman numerals are sequential but definitely not isochronous. The three ages of the Chaining of Melkor were obviously vastly longer than the three periods between the poisoning of the Two Trees and the first rising of the Sun, ‘but of bliss and glad life there is little to be said, before it ends’.
The second problem has to do with the beginning of the Years of the Sun. I have assumed that FA 1 (or, more accurately, YS 1) began with the first rising of the Sun in the West. But the reform by which Varda returned night to Arda soon changed the direction of movement, and by the end of FA/YS 1 the Sun was probably moving from East to West. Since this realignment took some time to accomplish, FA/YS 1 probably lasted more than 365 days. If, however, the Years of the Sun should be counted from the first rising of the Sun in the East, then all these dates may need to be adjusted.
Incidentally, I have assumed that the year begins in the spring, following the example of the Eldarin loa in providing dates for these Eldarin accounts of the First Age. Thus, the birth of Tuor, which probably occurred in January (Rían conceived two months before Huor went to the Nirnaeth Arnoediad at Midsummer), would be dated 473 in the Calendar of Imladris, but 474 in the Kings’ Reckoning. Of course, if the birth of Nienor ‘in the first beginning of the year’ occurred in January instead of in March (both are possible), then her birth would remain dated 474, but Dagor Bragollach should be changed to 456 and the Fell Winter to 497.
The greatest difficulty, though, involves the establishment of firm dates for most events. The great majority of the references to time in Quenta Silmarillion date events in terms of time elapsed since other events. Unfortunately, these datings occur infrequently, and out of all of them no more than four refer to any single common event: Mereth Aderthad is held ‘when twenty years of the Sun had passed’; Dagor Bragollach begins in winter, ‘it being then four hundred years and five and fifty since the coming of Fingolfin’; Nargothrond falls in the same year that the messengers of Círdan deliver to Orodreth the warning of Ulmo, ‘when four hundred and ninety-five years had passed since the rising of the Moon, in the spring of the year’; Eärendil is born in the spring, ‘five hundred years and three since the coming of the Noldor to Middle-earth.’[5] It is clear that Fëanor came to Losgar some time before Fingolfin crossed the Helcaraxë. The Moon rose as Fingolfin first entered Middle-earth, and it had crossed the sky seven times (days? months?) when Fingolfin entered Mithrim at the first rising of the Sun. I have dated Mereth Aderthad as FA/YS 21, since one Year of the Sun had passed at the beginning of FA/YS 2. Similarly with the messengers of Círdan (496, i.e., the next spring) and the birth of Eärendil (504, nearly eight years after Tuor’s arrival in Gondolin). But I have dated Dagor Bragollach 455, assuming winter to occur at the end of the year and ‘the coming of Fingolfin’ to refer to his crossing of the Helcaraxë a number of months earlier than his arrival in Mithrim. (Incidentally, an indication of the considerable length of time involved in the return of the Noldor is that during this period Melkor returned to Middle-earth, quarreled with Ungoliant at Lammoth, rebuilt Angband, and overran Beleriand as far as Amon Ereb and the Falas.)
But even with this juggling of dates some problems remain. The events between Dagor Bragollach and the fall of Nargothrond can be dated fairly closely by examining the careers of Túrin (born in the year when Beren first saw Lúthien, and eight years old in the Year of Lamentation, and twentythree—sixteen years of youth, three of slavery, and four of outlawry—when he journeyed to Gondolin during the Fell Winter that followed the autumn in which Nargothrond was sacked.) This kind of narrative dating is somewhat unreliable; Tuor’s three years of slavery could easily be thirtyone months—or thirty-nine. It is doubtless because of this imprecision that the dates given for Dagor Bragollach and the sack of Nargothrond seem two years too close. To reconcile this, I have removed two years from Barahir’s outlawry in Dorthonion (according to the text, he is killed ‘in that time’[1] in which Galdor dies, seven years after Dagor Bragollach, i.e., about 462) and one from Tuor’s enslavement in Hithlum.
In short, I must emphasize that the dates in this Chronology (and in the Guide as a whole) for the First Age depend totally on my interpretations of information that does not really warrant this kind of scrutiny. But while there may be errors in the absolute values of the dates, they can nonetheless be relied on as an accurate indication of the relative sequence of events and life-spans. If some of my conclusions seem those of some wayward pupil of Findegil the King’s Writer, rather than of a careful scholar and Elf-friend like Bilbo Baggins, my only excuse is the poverty of this Age; I have been unable to ask the Wise to emend my eagerness with their knowledge.
[There is full chronology of the First Age from the creation of Eä to YS 543 in the original text, but I omitted it.]
NOTE: From this point dates must be figured backward from the end of the Age. Since Elros died in SA 442 at the age of 500, he was born 58 years before the end of the First Age. If Tuor began to feel old at age 70 (as I have assumed here), then the First Age ended after about 600 Years of the Sun. This is not to say, of course, that the First Age lasted six hundred years; the Sleep of Yavanna alone may have endured for the equivalent of tens of thousands of Years of the Sun.

So, the proper definition of those last six centuries is 'the Years of the Sun of the First Age' instead of 'the First Age of the Sun'. The latter term never occurred in original Tolkien's writings, by the way.