They were Greeks with Roman citizenship. In other words they were both Greeks (ethnicity) and Romans (politically) and that's how they saw themselves based on the primary sources that have survived.

>When you today go to Southern Cyprus and ask the people, they will tell you that they are Cypriots or Rums (Romans), but not Greeks.

I suppose you mean Cyprus and I think you got confused by our different ethnonyms. Greeks today have three ethnonyms: Ελληνας (Hellenas), Γραικός (Graikos) and Ρωμιός (Rhomios/Roman). All these terms mean "Greek" in Greek and Έλληνας is by far the most popular one. Although during the byzantine period we also used Ρωμαίος/Rhomaios/Roman, today Ρωμαίος is used for the ancient romans while Ρωμιός is the term that we use for ourselves. As for cypriots, we are Greeks. As far as I know Turks use the term "Rum" for Greeks from Cyprus,Pontic Greeks and Greeks that still live in turkey while they use the term "Yunan" (from Ίωνες, the ancient Greek tribe) for the rest of the Greeks that live in Greece.

Yes, but even in that case the vast majority of Greeks remained. There were some Greeks that fled to certain European regions but there wasn't a mass migration.