Trying to answer the question Why did the term "Byzantine Empire" enter common usage instead of "Eastern Roman Empire" or "Roman Empire", I find a presentation by the Greek Byzantologist Helene Ahrweiler at the France Culture radio. (The discussion is in French, accessible on the France culture website and on Youtube).
She says HERE that old Catholic anti-Byzantine bias is still reflected in the historiography of Catholic tradition by the use of the term "Byzantine", while those of Protestant tradition prefer "Late Roman Empire" (the "Anglo-Saxon") or "Oriental Roman Empire" (the Germans). I guess she means modern, contemporary historians, because some 19th century ones were nastier then Gibbon or Hegel.
In a comment under this meta answer I see this statement:
"On Wikipedia "Eastern Roman" redirects to "Byzantine Empire". I believe this [removing "Byzantine" and preferring "Roman"] corresponds better to the terminology used by a lot of Eastern-Europeans (the Greeks called their empire "Roman" until the end), but this website is in a Western European language." etc
Are the English-speaking and German-speaking historians prefering other terms instead of "Byzantine"?