Byzantine Empire Imperium Graecorum or Imperium Romanum??

[keravnos]

I consider them Romans and later on the Greek succssors of the Romans more than anything else, I consider the Franks to be something of the Germanic successors of the Western Romans.

Exactly. I consider them as east romani speaking greek. It was only after us Greeks were under Turkish occupation that we considered ourselves as Greeks. Prior to that we thought ourselves as Romans, and so did the Turks. "Rumi" they called us then, aka Romani.