Yes, one way of referring to Latin is "lingua Latina" or just "Latina", but there's an old custom of using adverbs to refer to use of languages. So Latine is "in Latin" or "Latinly" (and there are similar adverbs available for other languages).
Interestingly, the language adverbs are also used in a construction with scire (to know) or intellegere: "Latine scit" (he or she knows Latin), "Graece intellegit" (he or she understands Greek). In English we would definitely think of this as needing a direct object, but Latin allows it as an adverb, to understand "in a Greek way" (perhaps it would make sense to think of it as something like "in a Greek manner" or "from a Greek perspective").
yep totally understand, had four years of latin in the 80s, some Greek, and many more. it's interesting to see how an idea gets phrased slightly differently across even related languages, i have to admit.