Folk mainly existed back when there was an obvious distinction between "music" (what highly proficient composers and lyricists set their names to and expected you to play without notable variation) and "trash" (everything else that one might play, whistle, or sing - with whatever variations or alterations that came to mind). "folk" being a later synonym for what was called either "trash" or "that nice song she's always humming," depending on your point of view.

Apparently this golden age, never stable, began to collapse between about 1850 and 1900, when signed compositions became available as sheet music at lower and lower prices; the bottom fell out when the phonograph became widespread after 1900.

At that point anyone could listen to whatever sort of music they wanted. "Folk" became the default designation for really old stuff performed by really old people in a really old style.

This really old stuff was treasured and revived by those of a conservative antiquarian, anthropological, and/or radical Marxist turn of mind. By 1960, it was obvious that whizbang arrangements of old songs and tunes could make a fortune for interested parties. That was when "folk" became a marketing label that came to mean (mostly) new stuff, not jazz, blues, or rock, performed by young white people with guitars. If said new stuff was in lyrical, pensive, confessional, modernist-poetic, or protesting mode, so much the better! A blend of all the above was better yet!

As a result, graying, desperate antiquarians and Marxists had to cling to the "traditonal" moniker for old anonymous stuff that the general public could stand only in small doses, if then. Soon rap music appeared, proving that anyone who could rhyme and had access to a turntable could get in the running to make a fortune. It was folklike in expressing popular, folky interests like love, sex, and violence, unfolklike in that only a philistine would perform somebody else's rap, thus short-circuiting the likelihood of a "traditional rap song."   

So in answer to your question: yes, "folk" exists, but in two forms. For most consumers it's the pensive guitar-type stuff mentioned above. For us pointy-headed geezers, it's what's there is when you can credit words and music to "Unknown or Irrelevant," and when you can also say, "Perform in any style you want, modify ad lib, don't worry about complaints from ASCAP."   

So it seems to me. And now, back under my rock!