A while ago I assembled a list of previous answers on the perception of Ancient Rome in the Middle Ages, which I can repost here below:

Most recently, u/J-Force and u/qed1 (with myself asking some follow-up questions further down) discussed the mediaeval perception of ancient history and what sources they would have used in this thread. Our u/sunagainstgold (here) and u/toldinstone (here) have written about what people in the Middle Ages thought of the city of Rome and its ruins. The latter of these has also discussed the Byzantine view of their past in these two answers, and in this one where u/WelfOnTheShelf also makes a great contribution. Furthermore there are two older threads in our FAQ that are also worth reading, this by u/qed1 and this by u/bitparity.

From the quotes in these answers you can see that they often just talked about "Rome" or "the Romans" (though someone more versed in mediaeval sources and philology than me could perhaps explain it in more detail). Our WelfOnTheShelf has also written about how people in the Middle Ages handled the naming of the Eastern Roman Empire here.

But even in Antiquity the terminology was not always as clear-cut as we might expect today, as u/bigfridge224 explains here and u/XenophonTheAthenian does here