"a capella" implies a certain kind of formality which unaccompanied does not. To call some kinds of singing and some kinds of songs "a capella" just because they don't have instrumental accompaniment borders on the ridiculous.
I agree. But some songs, particularly some versions of ballads, have the kind of dignity and formality that, at least to me, make the term "a capella" seem perfectly appropriate.
I have a hard time thinking of sea chanteys, for example, or a whole bunch of other songs sung without accompaniment as being sung "a capella." In fact, a sea chantey sung without accompaniment is closer to its natural habitat than it is when it's accompanied by an instrument (with the possible exception of a fo'c'sle chantey accompanied by, say, a concertina). The deck of a ship when raising sail, for example, is not exactly a chapel. But there are certain similarities: in a monastery chapel service there was often a line sung by a single monk or a group of monks, followed by another line sung by the rest of the monks. Call and response, or "antiphony," sung "a capella." With no instrumental accompaniment and literally "in the chapel." No harmony early on, however. Polyphony gradually evolved step by step over the centuries and morphed into what we now call "harmony."
For more than anyone could ever possibly want to know on the subject of liturgical singing and chant, check HERE