So I think your problem is that you don't directly describe what it is to be 'Roman' because the term itself it not a very concrete term.

Romulus was a Roman, Augustus was a Roman, Septimius Severus was a Roman, a low wage worker in Constantinople in 450 was also Roman.

People and identities change over time. They are allowed too they adapt. But if you asked every one at everytime what it means to be a Roman, the answer is different.

To Romulus a Roman would have been a citizen of Rome, by the time of Augustus an Italian would have been a Roman. Something unthinkable even 50 years prior to his birth. The Romans literally went to war with the other cities of Rome during the social war to finally decide once and for the question of what it means to be a Roman citizen.

Eventually you have emperors who aren't even born in Italy, Trajan was from Hispania and his ties to the Italians are sketchy at best. Yet he was known as the greatest emperor, Optimus Princeps.

We use Byzantine because it's easier for us as historians to have that need dividing line between the Latin and the Greek empires, but it's an arbitrary line in the sand. It doesn't make the Byzantines any less Roman themselves. It just makes it easier for us.