Which enemies of Rome are seen as barbarians?

[Lyceum]

Is that a serious opinion?

I'm not sure whether it was designed specifically at the Persians originally. But the Persians were the second nearest major group of non-Greeks to what is now modern Greece (after Illyrians, Thracians, and possibly Macedonians if you really want to stir up a storm in the modern political climate!), and to the Greeks of Asia Minor, the Persians and their vassals/conquered people were the closest group of non-Greeks. Considering it's people like Herodotus of Halicarnassos (in Asia Minor) who begin writing History, I'd say answer the above question with a clear "yes".

By the way, does anyone know where the explanation that "barbaroi" means "non-Greek-speakers whose language sounds like "bar-bar-bar" mumblings" comes from?

This is the only thing I'm going to really comment on here. Basically its onomatopoeic, like quite a few words in quite a few languages. You can find a few parallels in Greek for such words such as the use of Klange by Homer during Iliad 1 somewhere describing the sound of the arrows, or some such.

If you want an Indo-European parallel the word has a startlingly similar pairing in Sanskrit "barbaras" which just means a non (north) Indic speaker.

EDIT: Also, barbophones appears in the Iliad, I believe to describe either the Lycians or the Carians.