Races of the Source vs the First and those unique to each

[Nightsword]

The Elfin Bow in the loorbook talks about how there was once an tall and slender race knowns as "Elves" that the Allagans warred with and made extinct I think. It's not known if they were related to the modern Elezens.

At the very least, it looks like some racial names were shared between the Source and the 1st, but due to various reasons the names changed between shards. Makes sense since they all had the same origins until the Hydaelyn split the world

It was also confirmed via minions that Elezens, Miqo'te, and Lalafell were known by the equivalent FFXI race names, Elvaan, Mithra, and Tarutaru, at one point in their respective histories on the Source, so there are already examples of names changing in game.

[Enkidoh]

It was also confirmed via minions that Elezens, Miqo'te, and Lalafell were known by the equivalent FFXI race names, Elvaan, Mithra, and Tarutaru, at one point in their respective histories on the Source, so there are already examples of names changing in game.

Not necessarily; the descriptions for the mithra, tarutaru and elvaan minions in the minion guide had a very tongue-in-cheek tone (using terms like "supposed ancestor"), and hinted that might have just been a bit of fanservice referencing to FFXI without actually welding that to the lore, suggesting even in-universe that it was just a bit of joking around by a trolling Sharlayan scholar. Besides, hyur in the First might be known as humes like FFXI, but the other races there did not share FFXI names (mystel instead of mithra for instance).

So until SE officially confirms that the races of people in FFXIV originally had the FFXI race names, I'm taking that as just yet another callback to a past FF game without specifically making it part of the lore.