The name 'Byzantine Empire' is actually a lie. No one ever lived in a country called the Byzantine Empire. The Romanoi lived in the Roman Empire, its capital moved from (Old) Rome to New Rome, as Constantine called his city. The name was made up by 'Enlightenment' historians with an anti-Christian axe to grind, who wanted to separate the 'good' pagan Roman Empire from the 'bad' Christian Roman Empire by denying the later the name Roman.
Until the late 18th century, if you called a Greek speaking subject of the Ottoman Sultan a Greek (Hellene), he'd have punched you in the nose: Hellenes were pagans, he was a Christian and a Roman.
The Turks and Arabs remember this. We Orthodox Christians are called Rum in Arabic, and the Turkish name for the Patriarchate of Constantinople is "Rum Patrikhi", meaning Roman Patriarchate.
I know but I'm just using terms that all historians have used for the sake of common parlance.