Why did the term "Byzantine Empire" enter common usage instead of "Eastern Roman Empire" or "Roman Empire"? (or Imperium Graecorum?)
Wikipedia says that the term Byzantine wasn't used until 1557, and that pope Leo around 800 AD named Charlemagne "Roman Emperor" for political reasons. Note that the Eastern Roman Empire fell to the Turks in 1453, so this name change comes more than 100 years later, during the protestant reformation.
Wikipedia also says:
in the Islamic and Slavic worlds, where the Empire was more straightforwardly seen as the continuation of the Roman Empire. In the Islamic world, the Roman Empire was known primarily as Rûm. The name millet-i Rûm, or "Roman nation," was used by the Ottomans through the 20th century to refer to the former subjects of the Byzantine Empire, that is, the Orthodox Christian community within Ottoman realms.
How and why did the term Byzantine supplant other western European words for the Eastern Roman Empire? Did the Historians who did this have ulterior motives?
Was it Gibbon in particular who is responsible for this? What were his motives?