What I'm meaning to ask is: What are the technical names for things like the Southern Accent, California accent (or Western, if that's what it is), etc.?
I assume that regional accent differences have some sort of grouping terminology.
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First, caveats: the distinction in usage between ‘accent’ vs. ‘dialect’ vs. other terms can be somewhat tricky in academic work, but I suppose they’re nearly equivalent in normal speech. Also, I don’t think any naming scheme for US dialects is ‘official’.
There are different conventionalized options used by different linguists. One clear option is The Atlas of North American English (ANAE) (Labov, W., Ash, S., & Boberg, C. (2008). The atlas of North American English: Phonetics, phonology and sound change. Walter de Gruyter.). This book is described on Wikipedia at (https://en.wikipedia.org/wiki/The_Atlas_of_North_American_English); the page gives dialect group options like ‘The North’, ‘The South’.
There is a more detailed scheme, still based on ANAE, at a different Wikipedia location (https://en.wikipedia.org/wiki/North_American_English_regional_phonology#Classification_of_regional_accents). The ANAE is not perfect—see (Thomas, E. R. (2016). The Atlas of North American English and its impacts on approaches to dialect geography. Journal of Sociolinguistics, 20(4), 489-497.)—but it is a major, classic source.
In the interest of giving an actual answer here, and not only a link, here is a 6-way labeling used by Clopper et al. (Clopper, C. G., Pisoni, D. B., & De Jong, K. (2005). Acoustic characteristics of the vowel systems of six regional varieties of American English. The Journal of the Acoustical society of America, 118(3), 1661-1676.):
It is not clear in your question how small or specific you want the dialect groupings to be, so I’ll simply mention that some of the more famous local varieties have their own names (e.g. ‘Pittsburghese’), especially for non-technical use.