I know that, throughout their history, the Eastern romans called themselves "Romans". I believe that's what they were called by their eastern neighbours. But what did the western peoples call them? The franks, the goths?
And throughout the years, for as long as the Eastern Roman Empire was a political entity?
Romans. They were referred to as Romans by the Ottoman Empire (the successor state/conquerors) up until its dissolution in the 20th Century.
The Eastern Roman Empire was just the Roman Empire until a British revisionist perspective until in the 19th Century, during the peak of Pax Britannica and arguably linked an idea of "British exceptionalism" that was allowing them to have a monopoly on the world.
They didn't want to be seen as linked to the people they were colonising (during the 19th Century the Ottoman Empire was basically just a vassal state to the British Empire). So the distinction was made between Rome and Constantinople, where one did not really exist.
This allowed for the view that the British were the best of Rome, and allowed them to use things the Romans did, like "civis romanus sum", which became a big factor in the Don Pacifico affair.
This interpretation of the two being separate is now pretty much the orthodox view.