Forget all the jumbled mess, just know that the people of classical and medieval Greece to the time of mid 20th century never even used the word Byzantine. Now as to the correct term, it depends on the time period. The ancient Greeks around the time of Roman dominance would call themselves people of Hellenes. The Christian Greeks meanwhile would term themselves Romaioi, and if you used the term Hellenes, the Romaioi would hate you even if he acknowledge his pagan past=Hellenes because to a Christian Greek, Hellenes was a forbidden and pagan past. Since the Romaioi was extremely assertive of the fact that his empire was the true Roman Empire/Basileia Romaion, so yes, a Romaioi, considered himself Roman. Now why do we have the creation of different terms like Byzantine? Blame Western European politics and probably the pope who crowned Charlemange. Since Charlemage was supposed the new Roman Emperor of the [Holy] 'Roman Empire', however, what do you do when there was an already existing empire in the East? Simple, you discredit them and call them the Greek Empire. Soon, Western Europe was calling the Greeks a bunch of things, and one of them was Byzantine because it became popularized by the West. [Note, Byzantine was the equivalent of saying the Greek Empire].