For me, "true" folk music tends to be stuff that was repeatedly passed by word of mouth, whatever its origins.
If you find a handed-down Appalachian ballad, complete with misheard but preserved burden and some odd interpolated phrasings, it's "folk" to me whether or not you can find its origin in a broadside somewhere.
After all, everything HAD an origin; somebody first created "Barbara Allen" whether or not we know who that somebody was. But with or without a creator's name, it's gone from hand to hand, and evolved and looped and sprawled. It's folk now.
I even feel the same way about kids' songs. I've only heard one set of lyrics to "On Top of Spaghetti," and therefore suspect it of not being truly "folk." But no matter what the origin of "Great Green Gobs of Greasy-Grimy Gopher Guts" may have been, it's evolved by word of mouth into hundreds of forms. Can you actually find anyone who didn't go to the same camp or school as you who learned the exact same lyrics? I can't.