they call it natural horsemanship because your trying to emulate what the alpha stallion/mare does in a herd situation (a natural situation). whenever they ask something of another horse, they approach, the ears go back, they bite/kick. lots of other subtle body language involved in the encounter. and for whatever reason they try to keep this communication with eachother as subtle as possible; they'll only put their ears back and bite/kick if the beta horse hasn't responded well to the more subtle body language.
we can achieve the same things with our horses, so long as we go through that consistent sequence
of: subtle body language - maximum necessary force, and all shades of grey between those 2 extremes. and release the pressure when the desired result is achieved ("consistent sequence" in bold because the sequence must
be consistent, regardless of how well or how poorly the horse performs the desired response, the only thing that should vary is when the horse get's more understanding of what we desire, we will be able to start releasing earlier, as the horse will have responded earlier). everything else in all NH programs (and indeed most other non-NH programs, the only difference being terminology. NH's "suggest, ask, tell, promise" is just sugar coating "precue, cue, harder cue, enforcement") is just details and different ways you can aply this concept both on the ground and in the saddle.
realistically, NH and non-NH are objectively much the same, one just promotes better horsemanship (not guaranteed) by relying less on the use of increasingly harsh equipment. other than that the principles of pressure/release are extremely similar.
some good articles (not for the faint hearted): Reinforcement - Wikipedia, the free encyclopedia Operant conditioning - Wikipedia, the free encyclopedia
so basically: "Negative reinforcement
(Escape): occurs when a behavior (response) is followed by the removal of an aversive stimulus, thereby increasing that behavior's frequency. In the Skinner box experiment, negative reinforcement can be a loud noise continuously sounding inside the rat's cage until it engages in the target behavior, such as pressing a lever, upon which the loud noise is removed." (with horses, it's best to bring the "negative" on slowly, but as in the case with the rat, release whatever "negative" you choose (with the rat it was a noise, with a horse it may be legs/reins etc) as soon as the correct response is achieved, to develop a language, then once that language is clearly understood, progressively up the ante (asking for canter departs and sliding stops rather than just single transitions, or combine the 2 to create collection, if your into dressage rather than reining))
that, the ability to read a horse and an imagination is all you need to be a good NH (or non-NH) horseperson.
that was some elaborate bracketing i did up there.