I hate the term 'natural horsemanship' and never use it. I think it has just become a 'buzzword' for the marketing of systems that are supposed to help people with little horse experience and 'hands on' knowledge. It helps some people and it is of little help to many others.
The term tries to fool people into paying a lot of money for equipment and videos that is supposed to turn the novice into a person that can teach a horse to do what they want. They are trying to fool people into thinking what they are doing is 'natural' when absolutely nothing we do with horses is natural to them. The terminology is 100% BS as is the marketing that is only set up to part mostly middle aged women and idealistic little girls from their money.
If a person has learned how to 'teach' a horse, step by step, to accept each next new step without bullying, pain and coercion, that person is well on their way to becoming a true 'horseman' in my estimation. Teaching anything to a horse involves applying 'pressure' to elicit a response and 'releasing' that pressure when the horse does the right thing. 'Natural' begins and ends right there. This is how horses learn proper herd behavior and learn when people are not in the mix. Learning how to do this effectively to obtain the unnatural result WE WANT, is as close as any horsemanship comes to being natural.
If a person has learned to simply 'close' all of the 'wrong' doors and 'open' the door you want the horse to go through using 'pressure and release', you have set the horse up to do the right thing. The only reward the horse needs is the release of pressure.
I have always preferred to call my way of doing things "Common sense horsemanship" because that is exactly what it is.