Did the ancient inhabitants of Rome who spoke Latin refer to themselves as Latins?

Question

In Ancient Rome (8th century BC to 5th century AD), did the inhabitants use the terms Roman and Latin synonymously to describe themselves as a people?

Answer 1

When "Latins" was used as a demonym (Latini) by the Romans, it referred to the Latin-speaking inhabitants of Latium, the region in which Rome is situated (modern Lazio, though the Italian region is bigger than the original), often specifically in contrast with the inhabitants of Rome proper, and then usually as the people who lived in the region before the founding of Rome.
This is the case in e.g. Vergil's Aeneid, where the Latins, under king Latinus, are hostile to the Trojan protagonist Aeneas, whose descendants would—according to the story—go on to found Rome.

Answer 2

No. Latin was normally used only to describe the language used in Rome, although sometimes there were references to "Latin rights" or Latin religious practices that dated to early periods in Roman history. For example, at one point in Livy it says this:

Latinae feriae fuere ante diem tertium nonas Maias, in quibus quia in una hostia magistratus Lanuvinus precatus non erat populo Romano Quiritium, religioni fuit. (The Latin festival was held on the third day before the Nones of May, and a religious scruple arose because at the sacrifice of one victim the magistrate of Lanuvium had not prayed for the Roman people, the Quirites.)

"Quirite" means a citizen of Rome. In general, the Romans considered themselves to be multi-ethnic from an early period and citizenship did not depend on ethnicity. In fact, one of the accomplishments of the second king of Rome, Numa Pompilius, was celebrated as being the integration of different factions into a unified Roman citizenry. If you know Roman history, you know that Rome was founded out of many non-Latin peoples, such as Sabines.