I just call it “Byzantium” and I say the Turks are illegal invaders sitting on occupied Christian land. :-)
Strangely enough, the term Byzantium is false and was false.
It was coined by German scholars in the 1800s to refer to the eastern Roman empire as a way of differentiating them.
But the “Byzantines” called themselves simply Roman’s!!
And the Turks who conquered them, called them Roman’s. So too did the Bulgars etc.
A funny story is in 1919 when Greek soldiers took over a Greek speaking island that has been Ottoman and not freed by Greeks during their 1821 independence.
The island children came to watch the Greek soldiers and one of the kids shrieked “Hellenes” (which is what the greeks call themselves today and which was used in the ore 1493 world go refer to rural Greeks.
The Greek soldiers said “what do you mean, you are also Hellens and speak Hellenic”.
And the child said “No, we are Romans!”