While I agree with most of what has been posted, I think the matter of a horse enjoying or not enjoying touch is personal preference on the horse's behalf. I have met horses that totally shun human contact and others that seem to crave it, right now I own two that seem to be extreme cases of both ends of the spectrum, and I do think it effects the training methods that are suited for each horse.
My Fargo loves to rest his head in my hands and let me stroke his cheeks or forehead. He will stand with his head in my arms for as long as I want to keep up the contact. He presents areas of his body he wants rubbed and "asks" for scratching at times. He also enjoys just hanging out without touching, and on rare occasions will refuse my affections, but for the most part he seeks contact with me. He also really likes to have contact with the reins and behaves almost insecurely when I ride bridleless. He loves rubbing and petting as a reward.
Whereas Indiana, my retired pony never wants to be touched, he tolerates grooming and only seems to enjoy contact when he needs something scratched while bug season is at it's worst. With him I respect his desire to be left alone, even though for years I wished he would let me display my affections
. In his case he HATES bit contact(he was given to me because he had developed the habit of bucking whenever a rider touched
a rein.) The biggest turning point in being able to ride him safely was when the bridle came off, he responded far better to body cue and taps from a stick than to the reins. I was able to reintroduce the reins to him after we established body cues well enough that the reins were rarely needed in communication. I learned to leave Indi alone as a reward for him.
To me natural horsemanship also has to do with identifying the natural tendencies of each individual horse, as well as understanding them as a species.