When did the Eastern Roman Empire get the title Byzantine Empire and start being treated as a different entity when they themselves didn't see any break in continuity?

by Aetius3

Every article and TV documentary will refer to Istanbul as the capital of the Roman, Byzantine and Ottoman empires. "Byzantine Empire" would have been news to the people living in it. I've read that even the Greeks started calling themselves Romanoi. I fully understand why we tend to make the distinction but since the empire itself never made it, why do we and when did we start this doing this?

WelfOnTheShelf

We have some previous answers to similar questions that will hopefully be helpful:

Why is it called the Byzantine Empire when the city was known as Constantinople for centuries longer than it was called Byzantium? by me

and more recently When the term Byzantine empire was introduced? by u/AksiBashi