Byzantium = An ancient greek city founded by colonists from Megara in 667 BC named after their
King Byzas.
Byzantium is a latinization of
Byzantion.
While the Eastern Roman Empire existed it was not known as Byzantium or the Byzantine Empire. It was known simply as the Roman Empire (Greek: Basileia Rhomaion)
or Romania (Greek: Rhomanía)
to both its inhabitants and neighbors, as it was a direct continuation of the ancient Rome.
The Eastern Roman Empire was distinguished from ancient Rome in modern times. To put it simply, "
Byzantium" does not exist in a direct representation of the literal greek half of the Roman Empire insofar as the people living at that time were concerned.
For RP purposes I suggest the roleplayers begin to distinguish accordingly.
Or not.
source:
http://en.wikipedia.org/wiki/Byzantine_Empire