Since Byzantine Empire is a term that was invented after the actual collapse, what was called the Byzantine Empire by its contepomraries?
I've discussed this very recently in a different thread (https://old.reddit.com/r/AskHistorians/comments/9ctvn7/did_medievalrenaissance_era_greeks_disliked_being/) but the short answer is that they called themselves the Roman Empire and got annoyed at people who didn't call them that. In 968 this led to the collapse of peace negotiations between the Byzantines and the Holy Roman Empire over holdings in southern Italy and the resumption of military operations, a primary account of which can be found here: https://sourcebooks.fordham.edu/source/liudprand1.asp
Europeans tended to just call them 'the Greeks' - Odo of Deuil, the Gestra Francorum, William of Tyre, Otto of Freising all just call them Greeks.
The Byzantine Empire was the remnant of the Roman Empire. Different historians switch the names at different points in history. Some call them Byzantines around the reign of Justinian whilst others only change the name until after the Islamic Conquest severely reduced Roman power.