Old-style saddleseat before the adjustable stirrup bars came in in the '80s was closer to the classical dressage position. I have old catalogs that show only a 19" cutback saddle with a 4" cutback head for adults, and the park saddle, which was size 17"- 19" for adults.
Today, the adjustable stirrup bars created in a whole new artificial "sit-back" style, and eq riders are usually riding a 22" - 23" long saddle with the stirrup bars set all the way back. That is why you see chair-seat and also why you see 4"-5" of flap out in front of the rider's legs now.
I know I have seen a lot better and smoother equitation riders than that woman in the black suit in the video. Her legs were flapping all over the place, and her toes were out in front of here in a (imo) very unflattering picture. The Shively 2000 and MMX saddles are what the equitation riders use now, and they force the rider into a chair seat.
^ is Wing Commander from the '40s. Decent equitation position.
Also, some trainers and people who don't ride equitation classes have an entirely different style of pushing back as far as they can and sticking their feet out in front. But, they are not trying to be eq riders at all. Not the slightest bit. Many tall people say that sitting on their pockets like that gives them a more secure seat on a rambunctious or green horse. Trainer style:
I uploaded 3 saddle pics. The pinkish saddle on the bottom is a Barnsby Battaglia, the same tree/seat as the Shively 2000 and MMX. It's easy to see why, even with the stirrup bar set back all the way, you will always be in a chair-seat position. (I flipped that photo, so yeah, you're looking at the off side not the near side, for you sticklers, lol. I was in a hurry.)
The flatter seat (top right) is a L&R Louisville. And the deep seat (top left) is a Joseph Sterling Millennium 3000 deep seat and that seat type is also called "equitation seat." It's easy to see that either of those two saddles results in a more natural position. A saddle that has its balance point nearer the center, even if it is a saddleseat cutback, will allow the rider to align in the classical position, head, hips, heels.