I like what he says about working with the horse that is loose, and standoff-ish , in the pasture. And, the horse that is so worried he is looking 'outside' of the arena example, well, that is just exactly what I was taught, too.
But, I have to say, that walking along, right close to a horse that may have a more 'in your face' ,pushing on you way of being, . . h m m m . . . I think that can build even more such behavior. Especially if while hand grazing, you maintain enough tension on the rope that the grazing horse is in a state of a constant but slight pull on the rope. That can , in effect, teach him to ignore the rope, which I think is a negative thing.
And, matching steps with a horse you are leading could lead to a person speeding up or slowing down to match up with a horse that has a tendency to ignore his handler and run ahead, or lag behind, so again, you may be reinforcing bad habits.
@tinyliny, I haven't watched episode 2 of this yet besides a quick skim, but he has a separate video regarding that exact issue with pushiness/hand grazing. There, he talks about the difference between asking a horse to follow versus letting a horse lead, and making it clear to the horse when he expects each. He emphasizes it is *okay* to let a horse decide where they want to go... but only when the handler has given the "go ahead."
When he has a horse on a line, he either expects them to be a follower, and walks clearly and intentionally/lets them hit the end of the line if they're not paying attention, or makes it clear he is to follow them but never creates any tension in the rope. He makes it a huge point never to give mixed signals by asking a horse to do something, then let him pull to somewhere else/ignore the line, then ask him to do something again, then let him make the decision, etc. He never allows a horse to lean on a line, if they hit the end of the line he expects them to pay attention and do as he asks, but never pulls on a horse and doesn't follow it with something, nor does he allow prolonged direct contact/tension. Perhaps other people have different methods of hand grazing, but I don't keep any tension on the line, nor does he (4:50 in the video).
If you wanted I could go hunt down that video because I'm not fantastic at explaining things.