There is singing and there is singing.

A more classical approach would be pitch perfect, a minim lasting the length of a minim, a crotchet half that time etc. Of course, this does not mean that heartfelt emotion is ripped out, (feel the anguish in tragic opera!) but most classical approaches are about the uniform pitch & timbre rather than relating a good story with feeling.

I could possibly describe that better by suggesting you listen to the classical singer Andreas Scholl work his way through "English folk songs" on some of his albums. (Preview in iTunes store and you will know exactly what I mean about the voice as an instrument being pitch perfect but stripped of soul.)

I was classically trained on the violin, and if I played a piece by Vivaldi, my style, length of notes, even posture would be very different to playing Irish jigs & reels. I even tried (for my benefit) to distinguish between violin playing and fiddle playing. Sadly, after an accident, I play neither as I cannot get my wrist to twist round enough.

Folk on the other hand..... I am not a good singer in the instrument sense and if unaccompanied can wander into keys that don't exist. My guitar style is based on hearing a bass pedal note to stop my voice wandering off to the next village. But... people like what they hear, because it would appear I put "feeling and respect" to the words. (Other's views!) I live near Doncaster and can say that there are some fine singers around happy to get up in a pub and take a deep breath.

I do think that voice coaching would by definition veer to the former rather than the latter. Not a bad thing in itself and if your idea of folk is interpretation of Vaughan Williams, such singing would more than fit the bill.

Of course, being trained to use the voice as an instrument and then adapting that for a more "folk" style gets you everywhere. See Maddy Prior for details....