A horseman uses phases. He starts with a small signal ("the good deal" according to Buck Brannaman, phase 1 according to Pat Parelli), and if the horse does not react, he follows through until he is effective ("Do what it takes to be effective" says Brannaman, Parelli calls it phases 2,3 und 4). 
If you look at it from a scientific point of you you will be facing terms like negative or positive reinforcement but also punishment.
I have been wondering for quite some time if the phase 4, which can be ugly sometimes, equals a punishment of the horse. I finally found a quote that answered the question for me (though I don't know anymore who said it): "Offer the horse a place to go". We should show the horse where he is supposed to go, more precisely, what he is supposed to do. And we offer him the chance to do it right. 
In my eyes that is the difference between a punishment and phase 4: If I beat my horse - as a reaction to a behavior I don't like - I won't show him any alternative to his wrong behavior. The phase 4 on the contrary always aims at showing the horse "the place to go".

"Don't make assumptions" is a quote by Pat Parelli. Our horses are not supposed to make assumptions, for example starting a maneuver before we have asked for it, and it is our responsibility to not teach them to make assumptions.I think the quote works the other way round too: We humans too should not make assumptions but take what we find and work with it, meet the horse where he is at and not at the place we think he is. An old mare recently taught me that lesson. I wanted to train her body, move her around a bit to stretch her and help her stay flexible. I focussed so much on my goal that I noticed quite lately that the mare mentally and emotionally was not at all capable to follow my plan. I actually confused and scared her with my focus. That has taught me once more how important it is to observe the and feel for the horse, to find out where he the horse is at and to start from there - not from the point we are at or we think the horse might be.

When asked for something, a horse will either react or respond. If he reacts, he does it intuitively, spontaneously, whereas a response is linked to him understanding what the human wants.  As we want the horse to think his way trough a task, we want him to respond, not to react.
Often, a reaction is the first thing that will happen before a response can follow. The horse will react instinctively, out of fear or confusion because it doesn't know how to cope with the pressure we are applying. If we release the pressure at the right moment, we teach the horse how to make us go quiet again. He can learn and think - and that is what will turn his reaction into a response. It takes practice and a good timing on our part. The horse learns that we allow him to think, that it doesn't have to hurry and that we always ask for something he is able to give us. 

The horse industry offers a lot of devices, tiedowns and bits that are supposed to help if the horse gets out of control. The problem is: Mechanical tools address the symptoms like running off or bucking - but they do not cure or take away the cause for the horses displaced behavior: the fear. 
Horsemen like Warwick Schiller or Buck Brannaman say that it's not the reins that control the horse - the horse needs to control himself. At the very beginning even without a rider. Pat Parelli puts it like this: How should a horse be able to behave with a rider if it's not able to behave on its own? It's our job to teach the horse how to get control over his emotions and how to behave. We need to get to their brain - giving them confidence and taking away the fear. If we are successful we won't have to deal with behavioral issues anymore.