I think the biggest problem with the confusion regarding dominance when handling horses is that too many people confuse "dominance" with "domination". IMHO, those are 2 different things when it comes to handling horses. Yes, they may have the same root definition according to the dictionary, but in action, they are not the same.
I can show dominance over a horse with a firm word or a dirty look. Dominance just means that you are the alpha over them. Depending on the individual horse, you can achieve that with steady leadership and a kind word...or you may be required to punish or physically dominate them to maintain the level of respect needed.
IMHO, to actually dominate a horse means that you use a level of force beyond that which is really needed to "break" the horse instead of "teaching" the horse.
In my mind, I don't or haven't, differentiated the 2 words that much. When I say I dominate Honey every chance I get, I mean I show dominance, not that I'm trying to dominate or break her will. For me, that kind of domination would take all the fun out of figuring her out and it would change her personality and I LOVE her kind of gruff/grumpy/boss mare/testing/exploring options kind of thinking. That indicates a very intelligent horse, to me.
If she's invades my space I send her out. No anger, no meanness, just, "Nope, you can't come in here, out you go" and move on. If she ups the ante by giving me the 'mare look' and/or crowding more then I get a little stronger but if she backs away and licks and chews, then we're good and it's over.
When her foal was first born, she snaked her neck at me and then went a few feet away and hiked her heels at me. I sent her around the yard until she put her eyes on me and 'asked' to come in and kept her look pleasant. Like I said before, I don't beat on her or nit pick her, I just make sure she understands where the lines are and what happens when she crosses them.
I've gotten the feeling that she wasn't handled/loved on much, just groomed, exercised, shown, bred and back to her stall; not much human interaction. She's a big mare, and has a very strong personality and I think she got away with bluffing her previous owner (s) because her "mare ears", "blue eyed death stare" is more of a habit than actual threat. You come toward her, she puts those ears back and looks unpleasant, but walk up and give her a pat or scritch and she tolerates it and doesn't act out. We're working on changing that habit by having a handful of grain every few times we approach her, she can't have the grain unless she gives us 'pretty ears' and looks right at us with a pleasant expression. She's starting to anticipate the reward by looking at us with a nice expression a little more often than she makes the mean look. She's started to nicker a little when she sees me and she'll now approach and stop and wait for an invitation, sometimes but not all the time yet, so she's getting it.
Right now she's one of those all business horses but she's learning to like people and to look forward to what we might do with her or bring her. Her colt is very friendly and enjoys his scritches and will come up to see what you're doing just because he likes people. We're hoping she'll come around to that point of view more as she lives here longer.