I could totally picture an Eastern Roman scholar purposefully referring to it as Latin and not Roman because, after all, he considered himself Roman but spoke Greek.
No, most people outside Italy didn't have Roman citizenship until 212, and people in the eastern empire didn't start calling themselves Romans until late antiquity -- around the 5th century.
'Latin' was always the name of the language. That's because it's the language of the Latins -- the Latini, the people of Latium (modern Lazio). Early Rome secured its position thanks to the Latin League, essentially an anti-Etruscan confederation. By the 4th century BCE Rome was the centrepiece of the Latin League. The phrase 'Latin League' is a modern label, but phrases like latinitas ('latinity') and ius Latinum ('Latin law') in antiquity denoted citizenship rights within the league.
As time went on and the empire expanded, the Romans granted or withheld Latin rights to exert political control outside Latium. All of Italy had Latin rights in the 1st century BCE -- but Italy was only a fraction of the empire. Roman colonies outside Italy were also given citizenship.
Latin was the language that went with Latin rights -- hence the name -- until all people within the empire (including non-Latin speakers) were made citizens in 212 CE.