The most common form of expressing "Latin" when referring to the language is, as you note, to use lingua latina. Here are a few examples, many taken from the Lewis & Short entry for lingua:
ita sentio et saepe disserui, Latinam linguam non modo non inopem, ut vulgo putarent, sed locupletiorem etiam esse quam Graecam. (Cicero, de Finibus Bonorum et Malorum, 1, 3)
I feel and often have stated that the Latin Linguage not only is not poor, as it is often popularly reputed, but also is even richer than Greek.
Quod quidem Latina lingua sic observat... (Cicero, Orator, 44, 150)
Cicero also provides us with many examples of sermo Latinus, e.g.:
cui (Catulo) non solum nos Latini sermonis, sed etiam Graeci ipsi solent suae linguae subtilitatem elegantiamque concedere (Cicero, de Oratore, 2, 7, 28)
Not only do we accept [Catullus] as a subtle and elegant master of the Latin language, but even the Greeks often accept him as such for their own language.
in Latino sermone (Cicero, de Oratore, 3, 11, 42)
Another common word used to mean in Latin--as is often meant when we say Latin in English--is the adverb latine. Thus, to speak Latin is often rendered as latine loqui.
Cicero's de Finibus Bonorum et Malorum contains several uses of the adjective latinus, -a, -um referring to the language alone. This excerpt contains three uses of "Latinus" as well as (for kicks) a more colloquial way of referring to Latin as the "mother tongue" or "native language":
Iis igitur est difficilius satis facere, qui se Latina scripta dicunt contemnere. in quibus hoc primum est in quo admirer, cur in gravissimis rebus non delectet eos sermo patrius, cum idem fabellas Latinas ad verbum e Graecis expressas non inviti legant. quis enim tam inimicus paene nomini Romano est, qui Ennii Medeam aut Antiopam Pacuvii spernat aut reiciat, quod se isdem Euripidis fabulis delectari dicat, Latinas litteras oderit? (Cic. Fin. 1, 2)
It is therefore rather difficult to satisfy those who claim to despise Latin writings. In such people I rather wonder, for one thing, why their native tongue does not please them in grave matters, when they are not unwilling to read Latin fables translated literally from Greek ones. For who is so hostile to the Roman name that he would despise and reject the Medea of Ennius or the Antiopa of Pacuvius while claiming to delight in the fables of Euripides?
(my rough translation)
As James noted in his answer, this adjective latinus can be used as a substantive as well, in the neuter.
Latinitas is translated as "pure Latin style, Latinity" in its L&S entry and has a more precise usage that means much more than "Latin." See the above entry for examples of its use.