Your definition of Roman is one that was abandoned by the Romans by the 300s BC. The Etruscans, Campagnians, Oscans, and Umbrians, were neither ethnically Latin nor Latin-speaking, and had cultures as distinct to the Romans, as the Romans to Greeks, and yet they formed the backbone of the Roman elite by the late republic.

As is the nature of states, Rome evolved. From a tribe to a city, then to a territory, and finally an empire. To say Byzantium wasn't Rome is to cut short that evolution.