Most of the songs that we think of as traditional songs are usually attributable to an author/composer. One definition of a folk song is that it is a variant of an originally composed song. Dan Emmett, for example, wrote Old Dan Tucker and Dixie.
Stephen C. Foster write Angelina Baker which became Angline the Baker fiddle tune.
Maud Irving and Joseph Fillbrick Webster wrote Wildwood Flower before AP Carter changed it a bit. Woody Guthrie never wrote an original tune, nor did Dylan. They took supposedly traditional tunes for their texts. The tunes probably had composers too.

The "folk style" just means that the songs were played without amplification on acoustic instruments or were written on acoustic guitar. Remember that Old Dan Tucker, Hoosen Johnny, and even Barbara Allen (which resurfaced in print) were the pop songs of their day. Dixie certainly was. And I used to think Wildwood Fla-er was the national anthem of the South in the Fifties.

We Shall Overcome could fit the category of a folk song and to say it was popular is an understatement.

What we call folk songs of the future will probably be the Beatles' songs or maybe Elvis's. Or Dylan's.

There are so many songs written in the folk style such as by Paxton, Phillips, Ochs, Woody, Pete and the list goes on without being so-called "traditional". Elizabeth Cotton wrote Freight Train and Gussie L. Davis wrote Goodnight Irene.

The deal with folk and pop is that it's sometimes hard to separate them. What they have in common is that they are both accessible.