I can't help much with the bridle trail hacck..I am guessing that is another name for English Pleasure but I have never heard the term as tail hack.
First a definition :). A "hunter" or hunt horse, isn't a type of horse but a particular way of movement. The hunter idea came from field hunters...fox hunters. In order to run a "hunt" all day, horses need to be able to minimize movement of the legs..in other words, they work long and low strided, not picking up their knees too high and not overjumping fences..they conserve energy with efficient movement. Unlike a jumper who gets over a fence anyway possible and may jump very high over a top rail, a hunter tends to clear them "lower" with less air between them and the jump. A horse working on the flat will deplete its energy very quickly if they have a lot of high action in the knees..the movement needs to be from the shoulder to conserve energy.
Now, the Hunter Under Saddle is a class that judges the horse's movement and suitability as a hunter. The judge is looking for a horse that has a long, relatively low frame. They should have a long, ground-covering stride and very little knee action. The knees should not bend very high but the horse should move forward freely from the shoulder to maximize the amount of ground covered, without looking rushed. The horse should bring its hind legs well under its body to push itself forward (the hind foot should fall in the track of the front foot) and should appear to carry more of its weight on its hind end than on its front end. The head should be carried in a relaxed manner with the poll slightly above the height of the withers and flexed so that the nose is slightly in front of a straight line drawn from the poll perpendicular with the ground. They don't need to be on the bit per se as you might see in a dressage horse but be more relaxed in the frame with the nose slightly ahead of the vertical position.
In Hunter equitation the rider's position, equitation, is mostly judged but the suitability of the horse as a hunter to the rider is also a factor.
If the bridle hack is a pleasure class, the horse is normally ridden with a slightly longer rein and the rider's position may be slightly more forward to give the picture that the horse is 100% bombproof and could go all day and that the rider trusts them implicitly. It is a class to judge the horse as the old version of the park friendly, go everywhere while the rider daydreams on their back and don't try to kill anyone or anything in the meantime :) It is a horse on auto-pilot.