[alephnerd]

Same way a large number of European and Islamic empires continued to use some variation of Caesar to mean King or Emperor (eg. Tsar in Slavic cultures, Kaiser in Germanic cultures, Geysar/Qeysar/Kesar in Indo-Iranian and Ottoman culture)

[boomboomsubban]

I've always attributed that to the full Roman empire, though it probably was both, as I was under the impression Basileus was more commonly used when speaking Greek like the Byzantines did.

I think this comes from my modern perspective, where I think "nobody considers the Byzantines 'Roman' though that's what they called themselves." Of course that wouldn't be true in 1000 AD.

[_glass]

Later they came to be called the Greeks mostly by the Latins, to kind of steal the legacy, i.e. you're not Roman, you're Greek. But for the Ottomans it was always Rum (Roman), even now it is more Istanbulite to call the "European" side Rum. Same as there is this story from the liberation wars, where Greek kids from an island came running to see the soldier. The soldiers asked, what are you looking for, and they say: "Greeks!". The soldiers say: "But you, too are Greeks". The kids answer, "No, we're just Romans."