In particular would a Muslim resident be called Rumi, or a Turk or a Seljuk? What identity would they perceive as most important? If they called themselves Rumi, would that create confusion when referring to the (Byzantine) Romans who would also be called Rumi?
Seljuq was a dynastic name. So a person wd not likely call herself that, esp as other Seljuq princes ruled in Persia. Turk is basically a linguistic term, contrasting notably with Arab/ic. It wd be an unusual context in which s.o. wd call themselves a Turk in the sense of Turkish-speaker. Rumi cd mean Anatolian geographically in Persian, Arabic and Turkish but as you note also a Roman/byzantine politically. Sometimes it doubled for Christian so that a Frenchman might be a Rumi. To answer your own question, ask: who is naming herself in conversation with whom and where are they having the conversation? If, as a Turkish Anatolian, i refer to myself when talking to a local Christian i am probably going to be "a Muslim". If in a political context, talking to a visiting subject of the Byzantine emperor, i might say i am a subject of the sultan of Konya .. That may be enough to start you off. :)