To me, "unaccompanied" singing would denote solo vocal-only performances, and might even be understood to include self-accompanied performances (e.g., guitar and vocal, as practiced by many of us).

Most of the music under discussion here involves group singing without instrumental accompaniment, where the singers are, essentially, accompanying each other. Hence, as far as I'm concerned, "a cap(p)ella" is a more accurate term than "unacompanied," even when the music is not "church style," regardless of the term's ancient etymology and its literal translation from the Latin (or Italian).

As noted throughout the thread, there are great examples of group vocals without instrumental accompaniment in many subgenres in and out of folk music, including sea-shanties, barbershop, jazz, etc.

Let me throw in a mention of the great neo-doo-wop group The Persuasions. "Doo-wop" was originally a grassroots urban "folk" form generally performed without instrumentation, right out on the proverbial streetcorner, with nonsense-syllable singing providing the harmonies, counterpoint, rhythms, etc., ordinarily supplied by bass, guitar, piano, horns, drums, etc. The popular records inspired by this musical movement generally put at least some instrumentation back into the mix ~ recording of purely vocal doo-wop didn't occur until a later self-consious "revival."

The Persuasions are the greatest example of this revival. Part of the doo-wop tradition is creation of new arrangements of already-well-known songs, and these guys have done a particularly outstanding job of that, recording full albums of their reinterpretations of reggae sings, Grateful Dead songs, Bob Dylan songs, etc.