More or less correct. Although it's maybe good to know that officially the Spartans had been at war with Persia for ages... (Sparte had been an ally of the Lydian king, and had agreed to help him in his war with Persia.)

About ideology: why do you think there's something called Europe? It's after all a Greek invention, to distinguish the free Greek/Western world from the Eastern and Southern world which was ruled by tyrants (the kings of Persia)...

Barabaros is a word to indicate anything that is not Greek, and from the Persian wars onwards it's main use is to name the Persians. It's said that this word is derived from the Greek idea that other peoples like the Persians or the Thracians didn't have a proper language - just a mix of indicipherable sounds. In Dutch, the word 'Gebrabbel' is a very good description of the Greek view on those sounds.