When did the Roman Empire stop being the Roman Empire

[Urselius]

Virtually any of the literary/historical works produced in the Byzantine Empire would be awash with references to "we Romans."

To my certain knowledge the following do:

Psellus - Chronographia (in Penguin classics as "Fourteen Byzantine Rulers")
Anna Comnena - Alexiad (also in Penguin classics)
John Cinnamus - Deeds of John and Manuel Comnenus
Niketas Choniates - translated as "O City of Byzantium"

The worst insult Frederick Barbarossa, as "Holy Roman Emperor," could think to call the Byzantine emperor, or in his and his peoples' estimation "the one true Emperor of the Romans," Manuel I was "King of the Greeks."