This is what I would like help and opinions on. Clearly horses and dogs are different. I don't really have an ethical problem with smacking at a giant animal that's biting me. I see many equestrians at my barn do it and they all have kind horses and good relationships. I just want reassurance that it doesn't cause more problems. Or, if people have theories or opinions on why smacking a horse seems to work and smacking a dog doesn't, that would be cool too. It's also possible that my instructor gave me awful advice. You can tell me why you think that, too :)
There is not one training method that is "right". And there are many different training methods.
I, myself, am in agreement with your trainer. Now, I do not hit my horses very often and only when they deserve it. But biting is one of those things. You can bet I am going to come unglued for about 3 seconds (make them think their world is about to end), and then go back to what I was going like nothing happened. *Note that the correction is applied immediately for this to work.*
Now, I'm not repeatedly beating on them with a baseball bat or anything of the sort. My little hand it going to do nothing to physically hurt a 1200+ pound animal. It's all a mental game.
My way is extremely effective. I've fixed nipping/biting horses near instantly with this. And my own horses don't bite. They know the rules.
Same goes with our dog (yellow lab). When the occasion calls for it, such as learning that food on a human plate on the coffee table is NOT for her, we'll give her one good smack. She's over 3 years old, and there's been maybe 8 times that we've hit her in that amount of time. She gets the connection. She's smart. And she's very well behaved.
One hit by my hand, again, is not going to physically hurt her. But it is one method to discipline a dog.
Yes, there are people who do not agree with hitting dogs and/or hitting horses. If a different method works for you, great! Do it. But my method is very effective too, and it works for me. There's a big difference between "abuse" and "discipline".