How did the Byzantines call themselves and their state?

Byzantium, as you know, was located in the east. This state was actually called Eastern Rome, because the inhabitants themselves called their empire in this way. In Greek, they called themselves Romans, that is, the Romans. The state itself was named the Romeian state. Historians have introduced a more convenient term Byzantium, since in Antiquity it was also used, only in colloquial speech. The capital was called New Rome, and not Constantinople, as many are used to. But this caused confusion among people.