Except for the Astral Era part, this is essentially correct - the above happened early in the Sixth Astral Era (which lasted nearly 1600 years).
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Even then the above series of events is from the Sixth Astral Era. The various movements and migrations of the various races in Eorzea becomes less and less certain the further back you go, to where anything prior to the Third Astral Era has very little information, and the First Astral Era has almost none available.
That being said, the Elezen do believe they were the first, but that's largely based on their status at the start of the Sixth Astral Era.