Did Greeks in the Ottoman age feel Greek or Roman? Why was Greek identity chosen and not Roman when fighting for independence?

Go to Names of the Greeks: much good information there.

On the eve of the Greek War of Independence, the prevalent term for Greeks was Roman (Romioi). That was what the simple folk used, and they used it to refer to Greek Orthodox Christians (the Rum Millet), as the folk of the East Roman (Byzantine) Empire.

The Westernising elite was starting to revive the notion of Hellenes, as heirs of the glories of Ancient Greece, rather than the shame of Ottoman rule—and Byzantium (not much more popular in the West than the Ottomans). From the Wikipedia article, the independence fighters themselves bought into the notion that they were fighting to become Hellenes: the Wikipedia article mentions that

General Makrygiannis tells of a priest who performed his duty in front of the “Romans” (civilians) but secretly spied on the “Hellenes” (fighters)

Makrygiannis—a barely literate peasant, but a gifted storyteller in his Memoirs—embraced his Hellenic identity; and once the Modern Greek State was established, the Hellenic identity was what Greeks were supposed to aspire to, and their Roman identity denigrated. The most touching instance of Makrygiannis’ embrace of a Hellenic identity was his account of how he came to own two ancient statues:

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I had two fine statues, a woman and a prince, intact—you could see the veins on them, that’s how perfect they were. Some soldiers had taken them and they were going to sell them to some Europeans, for a thousand thalers. I went over, I took the soldiers aside, and spoke to them. “These statues, even if they give you ten thousand thalers, don’t you stoop to letting them be taken out of our country. These are what we fought for. (I took 350 thalers out and handed it to them.) And when I reconcile with the Governor [Ioannis Kapodistrias], I’ll hand them over to him, and he’ll give you whatever you ask for, so they can stay in our country.” And I’d hidden the statues away. Then, with my report, I offered them to the King [Otto of Greece], so they might be of use to the country.

But in those same memoirs, Makrygiannis recounts that, on the very eve of the War, a Greek excitedly said, “What do you think? We’ll go to bed in Turkey, and wake up in Greece!” But he didn’t call Greece Hellas. He called it Romeiko, the Roman State.

There was a third word, Graikos, that is, of course, just Greek. Modern intellectuals have occasionally used it to differentiate Greek Orthodox Christians (including Slavs, Arvanites and Vlachs) from ethnic Greeks. But it was not used that often.