The premise of the question is off: in a very real sense, the Romans were Latins, and the Romans were just as much the original speakers of the language as anyone else.

First of all: yes, the Romans definitely identified their own language as Latin in the classical period. Latin was lingua Latina, Latin speech was Latinum, the quality of writing "good Latin" (that is, grammatical and stylish Latin like that of high-status speakers) was Latinitas. Links are to the lexicon of Lewis & Short, but if you'd prefer a primary source, Cicero De Officiis 1.133:

nihil fuit in Catulis, ut eos exquisito iudicio putares uti litterarum, quamquam erant litterati; sed et alii; hi autem optime uti lingua Latina putabantur.

"There was nothing about the two Catuli such that you might consider them to have any extraordinary literary taste, though they were well-read - but then, other people are too - yet these men were considered to use Latin most well." (My translation)

The area around Rome was Latium, and it was inhabited by the Latin peoples. Rome was simply the power that emerged as dominant in Latium, and Romans were simply those Latins who formed Rome and ultimately unified all of Latium (and much of the world), including people from different parts of Latium. Strictly, the conflicts between Romans and Latins were conflicts between Romans and other Latins. Insofar as these other Latins were later treated as "second-class citizens" (Roman citizenship is worthy of its own question), that didn't stop the Romans from identifying themselves with Latium and with the language Latin.

For a (very rough) analogy, consider Latin:Roman::European:British - Britain is in Europe, but British people still identify themselves as different from other Europeans, it would still make sense to talk about a war between Britain and Europe, etc.

Source: Mommsen, T. The History of Rome