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."