Ok cool but the east in this instance would lack the legitimacy of the position of president of the US as they lost continuity and were conquered. Meanwhile the west, even though it is younger, continuously has a president ever since the split. The ruling classes in the east force the leftovers of the congress of west to elect one of their own as a president to legitimize their takeover.

At which point exactly was Rome conquered by the Greek by that logic?

First off, “a few latin speakers” is what we have today. At that time there were so many Vlachs and Armonians that we had special laws to cover them. And they were all over the place from Croatia to the Black Sea and from Vojvodina till the south of Greece.

The “few” latin speakers boosted the populations of Slavic states and still managed to create Romania later on. That’s how few they were.

In the 12th century there are still quite a few latin speakers in Constantinople.

On top of that,

Since when are Greek and Roman culture separated to a degree that a change of language is to be considered a change of culture? As far as i know Romans and Greeks functioned in synergy for a long time.

The status of Roman heritage between Rome and Constantinople in other words Catholics and Orthodox is a political thing.

Serbs called them Romei, Bulgarians called the Romei, Bosnians called them Romei. This i know for sure.

Disagreed. They called each-other Roman, other peoples called them Roman, the change of language was gradual but latin speakers were always present and latin was till widely used, they are quite literally The Roman Empire evolving into the middle ages.

Byzantium is what historians named it.

If they call themselves Roman and others called them Roman in their own time and after why would scientists name them anything else aside from marking it as a different period of Rome?

It’s just a term we use to mark the period like the Republic, the Monarchy and the Empire.