Is it a dialect? Sub-dialect? Something else?
The Cambridge Grammar of the English Language by Huddleston et al. (2002) says on page 5:
The regional dialects of Standard English in the world today can be divided into two large families with regional and historical affinities. One contains standard educated Southern British English, henceforth abbreviated BrE, together with a variety of related dialects, including most of the varieties of English in Great Britain, Australia, New Zealand, South Africa, and most other places in the British Commonwealth. The second dialect family we will refer to as American English, henceforth AmE – it contains the dialects of the United States, Canada, and associated territories, from Hawaii and Alaska to eastern Canada.
Therefore, the terms American English and British English are defined as referring to groups ("families") of regional dialects of Standard English in the context of linguistics (and that book – currently the grammar of the English language). Note that that excludes non-standard dialects, despite the terms American and British – namely, there's a variety of non-standard dialects that exist alongside standard ones.
Further note again that the text is talking about families of dialects, not a single dialect. There are multiple different Standard English dialects the term American English encompasses. (The same is true for British English.)
A Dictionary of Linguistics and Phonetics by David Chrystal (2008) contains the following entry for dialect:
dialect (n.) A regionally or socially distinctive variety of language, identified by a particular set of words and grammatical structures. Spoken dialects are usually also associated with a distinctive pronunciation, or accent. Any language with a reasonably large number of speakers will develop dialects, especially if there are geographical barriers separating groups of people from each other, or if there are divisions of social class. One dialect may predominate as the official or standard form of the language, and this is the variety which may come to be written down.
The distinction between ‘dialect’ and ‘language’ seems obvious: dialects are subdivisions of languages. What linguistics (and especially sociolinguistics) has done is to point to the complexity of the relationship between these notions. It is usually said that people speak different languages when they do not understand each other. But the so-called ‘dialects’ of Chinese (Mandarin, Cantonese, etc.) are mutually unintelligible in their spoken form. (They do, however, share the same written language, which is the main reason why one talks of them as ‘dialects of Chinese’.) And the opposite situation occurs: Swedes, Norwegians and Danes are generally able to understand each other, but their separate histories, cultures, literatures and political structures warrant Swedish, Norwegian and Danish being referred to as different languages.
The first definition of 'dialect' I found in an online dictionary is: "a variety of a language that is distinguished from other varieties of the same language by features of phonology, grammar, and vocabulary, and by its use by a group of speakers who are set off from others geographically or socially", so American English could fit that definition (as could any national variety).
But note the first two words: "a variety" - I agree with David Siegel that the best words are "variety" or "national variety".