As a general rule, the concept of Romanity had existed among the Greeks since the pre-Byzantine period. Roman policy meant that Greeks generally saw themselves as equal partners in the Roman state. For example, the Edict of Caracalla granted Roman citizenship to nearly all of it's subjects, which included nearly all Greek speakers (minus a few who had long since left to colonize Iran). This became truly cemented in the early Byzantine period when the Roman state hellenized in it's entirety. Pagan and Hellenos were seen as something of dirty words, meaning that one was backwards. Nearly all Greeks at this time adapted the name, Rhomaios, save a few stragglers in Greece itself (long since relegated to being the backwater of the Hellenic world).

The post-division Roman Empire of the middle ages, commonly referred to as Byzantine, saw one more fundamental change. During Late Antiquity, nearly all Greeks also became Orthodox Christians. This meant that Christianity and Romanitas also became fundamentally intertwined.

The Byzantine Empire hellenized even further under the reign of Flavius Heraclius Augustus (Ιρακλιος), who phased out all of the old Latin titles and replaced most government terms with Greek equivalents.

This didn't change after the Byzantine period, for two reasons:

1. Such a drastic cultural change fundamentally tied into the religion is a bitch to uproot

2. The Muslims, having grown up in the shadow of the Byzantines, largely adopted their view that Roman = [Orthodox] Christian

In fact, the Ottomans were keen to cultivate the Roman identity, largely as a counterweight against the remnants of the Crusader States, and any Papal effort to convert the Greeks. Thus, the Sultan took the Rum Ortodoks Patrikhanesi under his wing, and the Patriarch of Constantinople became the intermediary between the Sultan and his Roman subjects.

It is important to note, that Roman referred exclusively to a Christian, and thus a Roman Muslim was a contradiction. A Greek speaker who converted to Islam was thought of as having "turned Turk", as a Turk who spoke Romeyka (the language of the Romans).

It is also important to note that Roman referred to more than just the Greeks. It was the Ottoman shorthand for nearly any Christian, from a lowly Bulgarian herder all the way to Queen Victoria.

The fundamental sea change back to Hellenos largely was a result of the nature of the Greek state. It was founded by the west, and rid entirely on it's back. Part of gathering western support was playing up the theme park version of Greece, dropping cities like Thessaloniki and Mystras for Athens and Spartí. Still, the majority of Greeks would have considered themselves to be at best both Greek and Roman. The final nail in Romiosyne was the 1923 population exchange.

To get any insight into the personal convictions of Ottoman Greeks, I would highly recommend reading Patrick Leigh Fermor's memoirs of Mani and Roumeli. You will learn a lot about the nature of Greek identity.