Here are just a few possibilities in Hebrew (I don't speak Hebrew, so please forgive me--and let me know--if some are way off):
- אגודת - Indeed used with this meaning
- ציבור -?
- עם (Not the best)
- יהדות--Roughly "Jewry" (countable)--In frequent use with this meaning--see here
- ממוצא -- In frequent use, but lacks the sought degree of precision
- בית, insofar as בית יוסף -- Bet Yosef -- is a common metonymy for Sephardic practice and Sephardim as a group
- "Ancestry": שושלת ,יוחסין ,צאצאי -? . English: Nation, clan, race, ethnicity
- כלל -?
- קבוצת - Used moderately often, as here
- Nusach, דרך, minhag, mesorah, שרשרת. English: tradition, ritual, rite
..And here are the words I've found that come closest to being actual answers to my question:
עדה - Eida /edah, eidot -- To be found here and here, among many other places. In short, "eidot" is the answer to my question -- although Lital Levy interestingly notes (in Poetic Trespass: Writing between Hebrew and Arabic in Israel/Palestine) that in Modern Hebrew, the term is used almost exclusively to distinguish, and subalternate, Sephardi Jews, and refers only rarely to Ashkenazim. She notes further that "Modern Hebrew has no neutral words for race or ethnicity" (41, n. 85, and 82, n. 71)
"Shevet" -- שֵׁבֶט -- may be a misnomer, since it means "tribe," and there is only a sketchy correlation between the Tribes of Israel and the modern eidos. But it is in fairly frequent (mis)use for this purpose, as seen here and here.
For academic treatment of the lack of words in modern Hebrew to describe this divide, see Words and Stones: The Politics of Language and Identity in Israel
By Daniel Lefkowitz, pp. 15-16, 83-88, et passim.