What a tough question! The problem is, the requirements not only vary from class to class, they vary between types of shows. Hunter/English classes are different at hunter/jumper shows, stock breed shows like AQHA and APHA, and breed shows just to name a few.
What I can say definitively is that Huntseat Equitation is judged on the rider. Judges look at form as well as control of the horse (how well it responds with as little aid as possible as well as correct movement). Under saddle classes are judged on the horse. How well it moves, how relaxed it is, how well it responds.
Huntseat and saddleseat are completely different types of saddles and styles of riding. Saddleseat is typically used with gaited horses as well as Arabs and Saddlebreds. The saddle is very flat. The horses have very high head carriage and more upright movement and high knees. The riders have longer stirrups and a more upright position as well.
Huntseat is the style of riding you see when jumping (but classes are also on the flat). Horses should move out at a good 8-10 mph trot, stretching out with front and back legs so that the back legs at least meet the front hoof prints. Riders have a shorter stirrup length and more a forward position. In a lot of the stock breed shows, horses have a long and low carriage with loose reins. At hunter shows, the horses should be more collected with a higher headset (not as high as dressage) and a steady but light rein contact.
My experience is mostly with the hunter jumper style, so I can answer more specifically about requirements for those types of shows. However, I did also show 4-H, and the huntseat classes there had a mixture of the AQHA style and the hunter/jumper style. I can tell you that, at least in PA, the hunter/jumper style tended to win out over the AQHA style.
Hope this helps to at least get the conversation flowing.