The second problem is along the same lines, the concept of "reward"--do horses even think of it in our terms? I don't think so.
To me it's one of the most fascinating things about the human-equine relationship. We see in a narrow focus, through time; horses see nearly everything in a globe around them, in a compressed, strongly (but not absolutely) now-time. We expect food to be a "reward" because to us, predators, it is. Horses more or less expect food to be around them at all times. They graze and browse.
So what is this "reward"? I believe it's a shift in the whole circle of their now-environment: a time for pause, to taste and move the tongue, maybe to look around. They like the taste of the sweet; but we've all had horses who wouldn't take a treat. It's not a guaranteed reward.
I've been trying to work out WHY any of our training even works. I'm beginning to think it's because horses in general ENJOY the break, or mark, or poke, in their awareness; just the way we (or I, anyway) enjoy these moments when I'm entirely in One Time, or call it timelessness, and the sense I'm in connection with another species. Clicker training can be extremely effective, but like so many things, it's more effective when you allow it to happen, instead of trying to force it.
Let me clarify why any of this work - both traditional training and clicker training. They both function off of the same principals: operant conditioning. In fact, all behavior follows these rules and they are demonstrated scientifically over and over again.
Basically, operant conditioning says that we behave in certain ways because we expect a certain outcome to come of it. This was first demonstrated by B.F. Skinner with the "Skinner's Box" - he observed that a cat trapped in a box with a special "escape" device operated by a lever (if I remember correctly) would, by trying various methods, eventually figure out how to escape. The way to escape wasn't obvious to the cat, and the cat didn't make an immediate connection between pressing a lever and escaping. However, after a few trials, the cat did make the connection and pressed the lever in less time. Eventually, it pressed the lever as soon as it was put in the box, allowing it to escape. Even though it didn't necessarily make sense why pressing a lever would lead to escaping, the cat made a connection between a seemingly silly action and escaping, and so then began to do the action more with the expectation of escaping.
After further research, Skinner was able to demonstrate these same principles in other animals to teach them various skills. For instance, animals like pidgins and mice will press a lever or button to receive food. The theory of operational conditioning and subsequently behaviorism came four basic types of actions that can shape behavior:
Reinforcement: this is anything that causes a behavior to happen more often.
Type 1: positive (+) reinforcement means adding something that makes the behavior happen more often. This is clicker training, because the horse is given a treat (or other reward) when they do what we want.
Type 2: negative (-) reinforcement means taking something away and that makes the behavior happen more often. It is very important that you don't confuse the word "negative" to mean bad - think of it like subtracting something. This is traditional training - the pressure is removed, and the horse is "rewarded"
Punishment: this is anything that causes a behavior to happen less often
Type 1: Positive (+) punishment again means something is introduced, but it makes the behavior go away, such as a firm smack on the nose for nipping.
Type 2: Negative (-) punishment again means taking something away, but it makes the behavior go away. This doesn't have a clear application for the horse world as far as I can think of, but it's like getting the keys taken away for coming home too late.
As I stated above, ALL beings, from fish and lizards to birds, horses, and humans, are motivated by these rules. The only things that change are the motivator. Since food is a basic need, nearly all animals (it doesn't matter whether it's a predator or not) are motivated by it. If you'd like an example, I'm sure I can find a video of pigeons pecking frantically just to receive a piece of food (and they're not exactly "predators"). Sure, some animals are less motivated than others by food, but it generally works pretty well with horses. However, clicker training doesn't require food - it only requires that you pair a cue (like a click) with a reinforcement (whatever the horse will work for), and then use the cue to mark correct behavior (and generally follow it up with the reinforcement to make sure the cue keeps being associated with the reinforcement).