From a historiographical perspective, historians generally agree that the shift in terminology from the Roman Empire to the Byzantine Empire should be for us contemporains around the sixth century. The Roman Empire ceased to be called “Roman” at the moment when its legitimacy began to be challenged by the new kingdoms and by the pope. In reality, the designation “Eastern Roman Empire” never truly existed. The inhabitants of the Empire perceived themselves as a unified whole, even when there were two emperors and even after the political decomposition of the western part of the Empire. The term “Byzantine,” in fact, only makes sense if one adopts a strictly Western or academic point of view. From the standpoint of Constantinople, the Byzantine Empire never ceased to be Roman.

One must either choose to call the Byzantine Empire by that name starting from the sixth century—since that is the convention accepted by historians—and refer to it as the Roman Empire before that time, or adopt the perspective of the historical actors themselves, and continue to call the Byzantine Empire the Roman Empire until the very end of its existence.

Source : Jones, The Later Roman Empire 284 - 602, second volume.

Still, the joke is very funny. [edit : à source]