or so I hear

It's the Greeks the story is about for barbarian is in origin a Greek word βαρβαρος (barbaros). The etymology for Babylon with which I am most familiar is from the Akkadian babilu 'gate of the gods', the -ilu being cognate with Arabic ʔallāh and Hebrew ʔel both meaning 'god'. I am sure that the verb babble in English and other Germanic languages was influence by the story of the Tower of Babel.

On another note, there was another Babylon in Classical times: the Roman town of Babylon, in present-day Cairo, Egypt. The Arabic name comes from ʔal-qāhira. 'the victorious'.

Ceci n'est pas un seing.