Were all people from the Roman Empire considered "Romans", or just people from the city of Rome?

Being "Roman" and how Roman one was, were basically the key questions of the last few centuries of the republic.

Initially Roman meant being from the political territory of Rome. This would have included the city itself as well as the land for several miles in every direction. In the early republic most Romans would have lived near Rome, not in Rome.

Over the next couple of centuries Rome conquered Italy and as such brought in a lot of different peoples. Each group was treated distinctly. In one side you had Latins, closely related people who spoke the same language. They weren't initially Romans, but they were given a lot of rights including the right to marry Romans. After a while most Latins were Romans via marriage. The next group were allied Italian states. These were under Roman rule, and were granted broad rights within the Republic. But things like marriage rights, voting, and office holding were barred. A city like Capua would have its own magistrates who were Capuan, but couldn't stand for office in Rome. Then there were conquered peoples who gained no real rights at all.

It should be mentioned that these groups are only broad classifications. One allied city might well have a different set of rights and responsibilities. Likewise individuals were frequently given unique rights or even granted citizenship. A non-Roman aristocrat of some city might be granted citizenship for service to the Republic and from then on his family would be citizens.

It should also be pointed out that not all Roman citizens were equal. There were hard class distinctions. Your class could at times determine who you could marry, how you were treated in the courts, and even when and to what degree you could vote. If your family belonged to a less prestigious century, your vote counted for a lot less. Non-Romans awarded citizenship generally were placed into less prestigious centuries. But at least you were in. So by the second century BCE citizens peppered italy. Even if they were a minority. Many of these citizens had never seen Rome itself.

In 88 BCE the Social war came to an end and all free men of Italy were granted citizenship. By the early Empire most free people in Italy were citizens, with a peppering of officials, colonies, and granted families across the Mediterranean. Most people in the Roman Empire were not citizens. That finally changed in 212 CE, with the Edict of Caracalla in which all free men in the empire were granted citizenship. By that time citizenship had lost a lot of its luster and meaning. So mostly it just regularized. After that point citizen and subject are functionally synonymous.