To complicate the matter, Barbara Allen became known from its source, a printed version in an old songbook. That suggests at one time it was popular, a pop song.

I think the argument is over whether songs sung by popular artists such as the Beatles
can be construed as folk songs.

In time, with variants, I think they can. But they have to be taken up by enough people to produce cultural variants, by which I mean they find themselves in a monolithic culture changed from the original version. Rock and roll is not a monolithic folk culture but a
genre of music engendered by the music industry and used as a label in recording stores
to sell that brand of music.

A folk song is one in which many have taken it up and found variants of it.
A good example would by the song "La Paloma" written by a Spanish composer
and disseminated in different forms all over the world. No one can dispute its
popularity in its original form.