As I understand it, the Byzantine Empire is a historian's label. But I wondered if western Europeans wouldn't call it the Roman Empire, because to them that was the Holy Roman Empire. So did they call it the Eastern Roman Empire? The Greek Empire? Something else?
The generic term for them seems to have been "The Greeks", with the Eastern Emperor usually refered to as the Emperor of the Greeks if not by name, although at one point in Thietmar of Merseburg's Chronicon he is referred to as caesar augustus.
This is in Regino of Prüm and Adalbert of Magdeburg's Chronicles and Thietmar of Merseburg's Chronicon, all written from the early tenth to the early eleventh centuries.