Well, technically there is not one "correct" way of holding the reins. There are methods that are "more correct" and/or more effective than others. The way that I find least effective is what I call "farmer style". You see a lot of draft and recreational drivers having the rein come into the hands either over the index fingers with the "bight" (end of the reins) through the bottom. Sometimes, they can also come in through any one of the fingers. The reason this isn't my favorite choice is because it is very hard to shorten the reins without losing contact on the horse's mouth. That is another reason I am not fond of "hand holds" on the reins. You can't adjust the position of your hands on the reins at all.
I teach my students to use the "English riding" position, with the rein that comes into the hand under the pinky or ring finger. The reins can be easily adjusted this way without affecting the mouth so much as the bight of the rein can be grabbed with the opposite hand.
You can also use a similar position but with a "bridge" made out of the bight of the reins. This will help people learn to keep their hands closer together and at the right angle. IMO (and most of the rest of the carriage world) hands should be no more than 8-12" apart with the backs of the hands at the same angle as the withers or harness saddle. The thumbs should be used to "lock" the rein in position. The reason the hand is "tilted" is because that puts the whip into a "balanced" position in your hand that is MUCH more comfortable and effective than holding it straight up. You should not have to "hold" your whip persay, as that would get tiring. When your hand is tilted, the handle is kept falling down in front by your thumb and the "butt" of your hand, but you don't have to "grip" it. I learned here that many breed show exhibitors think that the judges want to see the whip straight up because that "means" they don't have to use it to get the horse to go. In carriage shows, the judges want to see you use the whip more so to get your horse to bend properly, raise their belly when necessary, lift a shoulder, etc. It "replaces" your riding legs.
Then there are also other very correct positions. See the recent Achenbach topic.
The reason you are seeing so many different rein holds at the breed shows is because Reinsmanship is usually not judged. The drivers are trying to draw attention to their horse with the driver's "showmanship", hence the overly exagerated height of the hands/arms, thumbs sticking up (I really don't know what the actual "purpose" of this would be. It seems like it would be uncomfortable and tiring for any length of time.
), and hands not level. With the bigger horses (Arabs, Morgans, etc.), the hands need to be high because the carts are so low, but really they don't need to be that high with minis. The performance of the horse is being judged, and the driver is really just a "passenger" trying to get the ultimate performance out of the horse. This type of "overshowing" is highly frowned upon in the carriage ring. You can still get an extremely brilliant performance with a "conservative" method. IMO, whatever method is chosen, it should be because the driver finds it most effective and logical, not because somebody else is doing it that way. Understand what you are doing and why.
You didn't know that you asked such a big question, did you!
Myrna