Thanks to AdamX, I've finally discovered Hot for Words, a very popular video blog by Marina, a Russian philologist who discusses word origins. I didn't know you could get a degree in philology, but she has two.
One of her latest episodes is on barbarian. As she says, it's from Greek βάρβαρος barbaros "foreign (esp. non-Greek-speaking), rude, prob. orig. referring to unintelligible speech" (ODEE). Specifically, βάρβαρος was borrowed into Latin as barbarus, and this is the source of barbarian and barbarous thru Old French.
βάρβαρος is related to Sanskrit बर्बर barbara "stammering" and Latin balbus "stammering, stuttering", all from Proto-Indo-European *baba- "barbaric speech". It seems it might have been the Indo-Europeans, and not the Greeks, who equated "foreign speech" with "bar bar bar".
At the end of this episode, Marina says that the Romans took the Greek word and applied it to another kind of person. I suppose she's referring to Latin barba "beard", but this has no relation to βάρβαρος. Latin barba is from Proto-Indo-European *bʰardʰ-ā- "beard" (also), and is the source of barber and barb (as in "sharp point"). beard is from the same root thru Proto-Germanic.
However, barb meaning "horse of a breed introduced by the Moors into Spain from northern Africa" is from Vulgar Latin *Barbaria "Barbary states" from βάρβαρος, according to the AHD. This implies that Barbary is from βάρβαρος as well. I cannot find any sources that convincingly connect βάρβαρος with Arabic بَرْبَر barbar "Berber".
I'm currently looking for a Russian blonde to host my blog.