No one plays guitar and sings songs by the likes of them.
Edith Piaf is part of the French Chanson tradition, which involves people playing guitars and singing the songs in informal settings, and in which songs pass around by oral transmission as much as anything. And chansons even bob up every now and again in English speaking folk circles. It's a genre that sort of overlaps with folk music proper.

It's premature to talk about the Beatles as "folk". In time, who knows? The songs are already in our common memory, in the same way as music hall songs are, and like music hall songs, thye fit well into the folk scene. In time I imagine they'll be collected in different versions from people who have never heard of the Beatles, and they'll have changed along the way. By that time it might be appropriate to class them as folk songs.