[YeGoblynQueenne]

I'm Greek and your information is correct :)

We have at least three words to refer to ourselves as a nation:

"Hellenes" is the one most commonly used in modern parlance. I'm not sure where it comes from but you could google it.

"Graikoi" I believe is from the same root as "Greek". You find it in literature from the 19th century and earlier.

"Romioi" is a rendition of "Romans" and you find it very often in accounts from the Epanastasis, the uprising against the Ottoman Empire in 1821. It's used by mainland Greeks and also by the Albanian people who had moved into Greece under the Ottomans, and then fought to expel the Ottomans alongside the Greeks, and finally are today considered Greeks (primarily because it would be madness to try to untangle one's heritage, after a few hundreds of years of intermarriage).

"Romioi" is not in common usage today, except in literature or generally as a colourful turn of phrase. I believe it was bequeathed to us from the Byzantine Empire, a.k.a. the Eastern Roman Empire, as others have noted.