While it is true that Byzantium was created by splitting the Roman Empire, Byzantium quickly evolved away from Roman traditions, forming a new culture that was uniquely their own.
The Byzantium’s were proud of their nation and did not consider themselves Romans, even though many of them descended from Roman families.
In time, the official language of the Byzantines became Greek, rather than the Latin which was used by the Romans. The Byzantines were Christians, but they did not follow the Catholic faith of the Romans. Instead they developed their own version of Christianity which eventually became known as Eastern, or Greek Orthodox.
Some folks never exaggerate--they just remember big.
The Ancient City of Rome was on the site of the present city of Rome, the capital of Italy.