Like I've said, I do agree this horse needs firmness, but I don't think that means it's *generally* best to use such strong methods as to cause pain. Mild discomfort and persistence until you get the smallest yield at what you want is IME just as effective with less 'side effects' IME.
I do agree with PP basic principles(effectively just negative reinforcement, but so saying, haven't been in the PP scene for at least 15 years, so I suspect there are a lot of changes), but understanding & applying the principles don't add up to the sort of 'recipe' approach & specifics others often use. As mentioned, I think it's not the principles but trying to follow a set of 'instructions' that tend to go wrong, because everyone has their own perspective, including the horses.
I too agree that 'clicker training' is a fantastic method for any sort of training, be that fearful or 'dominant' type horses, be it for 'tricks' or 'real work'. It's actually about my most effective 'tool' for changing motivation & behaviour of 'rude' or 'make my day' type horses. I think, as with most things, it is vital to understand the *principles* behind it though, and the specifics(eg. Whether you use a plastic clicker or otherwise, whether food treats or other +R) are optional.
I personally tend to use a combination of both positive and negative reinforcement - both trainer's methods you describe, without going to the extreme of a stud chain to 'pop' him painfully or such(unless it's a safety concern, which means I can't afford the patience to teach the horse), and while I use 'c/t' principles, I don't use a clicker & only part-time use food treats as reinforcement.