This is a very interesting thread - you really got me thinking!! Thank you :)
I've learned the term "punishment" to be defined as "something that causes a decrease in the frequency of the behavior" and "reinforcement" to be "something that causes an increase in the frequency of the behavior".
But I've never thought of the difference between a correction and a punishment. I suppose in modern language the term "punishment" is not as simple as it began, and has changed to encompass a more upsetting emotion behind it.
I suppose I see the difference being "punishment" - "something that causes a decrease in behavior" while I would define "correction" - "guiding them to the correct choice". Because when I correct my horse, who may be going faster than I asked by a gentle squeeze on the reins and a heavier seat. I may correct my horse for invading my space by backing them up firmly or yielding whatever body part invaded my space (hip or shoulder or otherwise). While I would reserve forms of (what I define as punishment) for things that are absolutely wrong, such as acts of aggression.
This brings me to the basics of learning, there are four way all creatures learn (please think of Positive and Negative like a math equation- adding and subtracting, not as in "good" or "bad"):
Positive Punishment: the addition of something unwanted to decrease the frequency of the behavior (so hitting a horse who bit)
Negative Punishment: The removal of something desired to decrease the frequency of the behavior (This is common with kids "no TV because you didn't do your homework")
Positive Reinforcement: The addition of something desired to increase the frequency of the behavior (giving a treat when the horse does the right thing)
Negative Reinforcement: The removal of something unwanted to increase the frequency of the behavior (most commonly used with horses, a release of pressure for the right action).
The trick to training any animal is balancing those four ways of learning and knowing when and how to apply each of them to get the best results.
I find many forms of "positive punishment" to not be very affective with horses, unless it was done to an extreme, as most often the action meant as punishment happens too late for the horse to connect the dots. And when we resort to only the use of positive punishment, we're really not fixing what caused the problem in the beginning. If the horse bit, there was a reason, hitting them may tell them "that's never allowed" but unless you fix what caused the initial trouble to begin with you won't get far.
I'd also like to add one more thing, I may be going too far off the OP, but I find the entire horse world really avoids Positive Reinforcement as a whole. Some people may use a pat or scratch to reward horses sometimes, but not consistently and that only works if the horse honestly appreciates it. My mare for example would do a back flip for a good belly scratch :P but other horses would be completely thrown off by a pat and may even consider it "punishment".
I find many people say "you can't feed horses treats, it makes them bad", but honestly, if you can't safely feed a treat, your horse isn't all that respectful of you is he? In using positive reinforcement training (clicker training or some equivalent) with horses it takes all of 3 minutes for most horses to figure out they only get food when they stand calmly and face forward, out of the human's space. Most horses I know who are clicker trained are fed treats all day every day and all of them know how to get treats, never mugging. So I'm still at a loss why the equine community has so completely rejected that entire form of learning? Perhaps it's because food is such a strong reinforcer and if not used correctly you could teach horses bad things? I work at a jumping barn where each horse has a terrible habit at feeding time, one grinds his teeth on the metal bars, several kick and bite their walls, many paw, a few pee - and each time they're fed while behaving this way the people are further reinforcing those terrible (and dangerous) habits. So if not use positive reinforcement for good things, I think people really ought to, at least, become aware of what they're actually reinforcing when they feed.
Thank you for giving me so much to think about, I'm eager to hear more responses and opinions on this particular matter. Sorry I went on, you really got me thinking!