I think it's actually fairly simple. The threshold between contemporary (singer-songwriter) folk and traditional folk is pretty recognisable to anyone who is gets really into traditional folk music - like most of the people on this forum.

Martin Carthy's treatment of certain pop and theatre songs (New York Mining Disaster, His Name was Andrew, The Wife of the Soldier) is a really good example. He clearly picks songs that have certain aspects in common with the folk tradition in their words and perhaps music, and that's presumably why he was attracted to them. He sings them with a delivery and arrangements that make it consonant with his other material.

Then there's the next degree of difference: singer-songwriters like the early Bob Dylan, Bert Jansch, Ralph McTell or Nick Drake, who are clearly very influenced by traditional music (who may even play a fair of traditional music) but who mostly play originals.

Then there's pop music that happens to be played on acoustic instruments (Mumford & Sons et al). We've all got ears here; you can call it folk if you like - most people do - but acoustic pop is probably a better term.

Isn't it just as simple as that?