I wouldn't call it a true chair seat, because it's not so much that your leg is slipping forward but that your seat is slipping back. You're sitting back on the cantle rather than the deepest part of the saddle.
In the last photo posted, it does not look like the saddle is level cantle to pommel, so I would experimenet with closed cell foam seat risers and see if putting a slight lift in the back of the saddle fixes part of the problem.
It also appears as if you ride with a hollow in your lower back, which has the effect of pushing your seatbones out behind you and distorts the hip/heel alignment. Once you get a riser for the back of the saddle and it's easier to stay in the deepest part of the saddle, work on relaxing the small of your back, and keeping your seat bones underneath you while you stretch up tall.