Okay, so I know the byzantines thought of themselves as romans, and called themselves so, as rhomanoi, but what about eastern europe? If I was in France in 800 or 1100 and I was talking about the ere would I typically refer to it as the byzantines? The Roman Empire? The Eastern Roman Empire? The Greek Empire? And how would I refer to the people? Would they be greeks, romans, byzantines or east romans?
"Byzantines" is a modern term, and the westerners considered themselves the heirs of the Roman empire. As such, they normally referred to them as the Greeks. For example, consider Einhard's references to the Byzantine emperors:
Imperatores etiam Constantinopolitani, Niciforus, Michahel et Leo, ultro amicitiam et societatem eius expetentes conplures ad eum misere legatos. (Vita Karoli Magni, 16)
A few years before this, Charles had asked him for an elephant, and he sent the only one that he had. The Emperors of Constantinople, Nicephorus [I 802-811], Michael [I, 811-813], and Leo [V, 813-820], made advances to Charles, and sought friendship and alliance with him by several embassies[.]
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Ex his omnibus duos tantum filios et unam filiam, priusquam moreretur, amisit, Karolum, qui natu maior erat, et Pippinum, quem regem Italiae praefecerat, et Hruodtrudem, quae filiarum eius primogenita et a Constantino Grecorum imperatore desponsata erat. (ibid., 19)
He only lost three of all his children before his death, two sons and one daughter, Charles, who was the eldest, Pepin, whom he had made King of Italy, and Hruodrud, his oldest daughter. whom he had betrothed to Constantine [VI, 780-802], Emperor of the Greeks.
Edit: Just to fully bookend the period in question, here is an excerpt from the Gesta Francorum:
Grecorum autem gens exibat et ueniebat gaudens in occursum domini Boamundi, nobis deferens maximum mercatum, ibique nostros tetendimus papiliones in quarta feria ante Cenam Domini; ibi etiam Boamundus totam gentem suam dimisit, perrexitque loqui cum imperatore Constantinopolim, ducens tamen secum paucos milites. (Gesta Francorum, 2.1)
The people of the Greeks came out, bringing us the greatest market, and went joyfully to meet Lord Bohemund. There we Pitched on, tents in the fourth day of the week before the feast of the Lord. There, also, the learned Bohemund left all his host and went on ahead to speak with the Emperor at Constantinople.