Amber was never a "kicker", but I have never treated her as incapable of doing so. Yeah, when I have horse-shy friends who are scared of her, I'll stand behind her and drum on her butt so it puts them at ease. But no matter how tolerant she is, a horse is a horse and a horse can kick.
I got her in December, so I did not have a chance to bathe her until the spring. we moved to the new barn in March and towards the end, it got warm enough that I started to hose her down with water that was slightly warm at the end of rides.
I'm fairly certain she's been bathed before in her life. But that didn't mean she liked it. She wanted to be especially nasty about having her hindquarters sprayed. So she lifted her leg and threatened me. And I immediately smacked her and growled at her. I acted exactly like the dominant horse in the pasture would. She threatened me, and I am the leader. A subordinate creature threatening me is offensive, so like the leader in the pasture (or wild for that matter), I "kicked" her for the offensive behavior.
She jumped, looked at me, and then everything went back to normal.
What a lot of people don't realize is, the seemingly little things we let them get away with - such as walking too far in front or behind us when we walk, nipping, being impatient, ect. All the little things that they get away with, that often we don't even notice or see as blatant disrespect - those things can lead to worse behavior. They think, "I got them on this...so maybe I can challenge them and become the leader!"
The key is punishment AND reward = correction. You can punish bad behavior, but that only leads to more bad behavior, or fearful behavior, because they don't know what the right behavior is. To correct a horse, you need to get the desired behavior and reward it, via pats, treats or verbal praise. Seems like your trainer knows this very well already.
To add though, the punishment should be quick and to the point. If a horse bites you, knock them in the nose with your knuckles. If a horse kicks at you, or threatens, smack them on the leg or hindquarters. Don't get angry or emotional, but absolutely make the horse think that you're going to murder them for what they just did. Then, when they behave nicely, praise and treat.
You probably already know this, but a spooking horse shouldn't be reprimanded, unless they are doing it purposefully. I rode a horse who was leased by a young girl and he did it on purpose just to get out of work. One good hard kick and he got his mind back on work. If he was really scared, he would have just dumped me and ran off - most horses will choose flight over fight!
Cinnamon Whiskey 11 y/o 15hh Chestnut AQHA mare, 2'6 Jumpers