To add to your comment: Anyone who didn't speak Greek was by definition a barbarian because that is what it meant. That is one was βαρβαρίζει (~varvarizi) (verb), as that is what other neighbouring languages sounded (and sound) in our ears: a constant βαρ βαρ (var var) sound.
The Romans added Latin to the non-sounding var var /barbarian languages, when they came in contact with us. Mostly out of prestige/common trade/allying. So the term was later enforced to mean anyone that doesn't speak Greek or Latin is a barbarian/"barbarizes."
And by the time of Pax Romana to mean inferior, as it did not have the knowledge and resources to learn the lingua franca of the times. In Hellenistic times, prior, the lingua franca was Greek, but I am not sure at the moment of the average Joe's perception on the matter. Generally a lot of trading cities where open to new ideas so I doubt that it was seen as inferiority more like a peculiarity or annoyance -- if you were trying to trade.
There is the notorious "to an unknown God" inscription/offering Athenians had [https://en.wikipedia.org/wiki/Unknown_God -- if you excuse my laziness with wikipedia]. That is not the behaviour of someone obsessed with themselves.
Hence the term 'barber' for a person that cuts said facial hair.
https://www.etymonline.com/word/barber
“barbarian” is onomatopoeic for “people who talk like bar bar bar bar and aren’t greek”. the (themselves barbarian) romans adopted the term.