In The Shaping of Western Civilization: From Antiquity to the Present, ed. Michael Burger (U of Toronto Press, 2014), we are given the following colorful detail (§2.3, p47):
What, according to the Greeks, made a Greek a Greek? The simplest definition in the Archaic period was that a person was a Greek if she or he spoke Greek. Hellenes spoke Greek and barbaroi did not. Indeed, Greeks believed that the term barbaroi itself originated in non-Greeks' inability to speak their language. To them, the speech of barbaroi sounded like nonsense: "bar, bar, bar..."---barbaroi.
And it's not just this book, either. The historian Philip Daileader also mentioned in one of his Great Courses lectures that the Greek word βάρβαρος has its origin in the 'bar-bar-bar' sound of foreign speech. And a simple Google search turns up other innumerable examples of authors (many with scholarly credentials) repeating the same.
But this just sounds very suspicious to me, like a modern myth. Hence my question:
(Q) Is there any verifiable ancient source for this claim about the word βάρβαρος (barbarian)?
EDIT: This question was closed on the grounds that its answer is contained in the Wikipedia link, but it is not.
To clarify, I am not looking for a scholarly reference, as I already have plenty of those. Rather, I am looking for primary sources from antiquity that supports the scholarly claim about the Greek word 'barbaros'.
In particular, the traditional explanation (that the Greeks invented this word because foreign speech sounded like "bar, bar..." to them) seems inconsistent with the existence of a Sanskrit cognate with the same meaning, which is also mentioned in the Wikipedia page. In this instance primary sources for the folk-etymology are critical, and Wikipedia does not cite any.