"speaks Italian better" vs "speaks better Italian"

As you recognized, both these sentences are grammatically correct, and have slightly different meanings.

The potential confusion here comes from the fact that "better" can act as either an adjective or an adverb. Once this is recognized, decomposing the sentences becomes simple:

Laura speaks Italian better than Jake

Here the subject is Laura, the verb is speaks, and better is an adverb modifying speaks. Thus, the thing which is better in this case is Laura's speech.

Laura speaks better Italian than Jake

Here the subject and verb are still Laura, but better is now an adjective modifying Italian (the object of the verb speaks). Thus, the thing which is better is the Italian spoken.

As the quality of spoken Italian is determined largely by the act of speaking it, there is obviously significant overlap between speaking well (when you speak Italian) and producing good Italian (when you speak it). As such, these sentences can be used interchangeably in most cases. There are cases where the second sentence, but not the first, would be accurate. For example, in a conversation between a Puerto Rican and a Spaniard, one party might assert that the Spanish spoken in their country is, as a language, superior to the other's. In this case, only the second form (Puerto Ricans speak better Spanish than Spaniards) would convey the intended meaning.