Regarding the chair position: If your seat and stirrups are in a line, then your 3 points of support are in a line perpendicular to the horse's spine. If you want to free up the horse's motion for maximum performance, that is good - and that is, I suspect, why it is right for dressage and jumping.
However, if your support is perpendicular to the horse's back, then sudden changes in the horse's velocity will have greater impact on your balance. If your horse suddenly stops, then you don't have anything to keep your weight above your points of support. Thus the cavalry recommended a chair position for the average rider and average horse, particularly covering rough terrain. IIRC, Dick Francis (of mystery writing fame, who was a famous steeplechase jockey before) recommended a mild chair position for steeplechase, and most western sports follow suit. Western and Australian saddles tend to force you into that position, although a western saddle does it in part to help with the impact when you rope - something most western riders never do. Sudden stops and sudden bursts of speed are easier to control with 3 points of support.
After sprawling a time or two on my horse's neck while riding an English saddle off property, I read the cavalry manual and figured it made sense. I freely confess I don't have enough experience to KNOW, and that this is just me trying to understand WHY various styles of riding exist.
On property, riding an English saddle, keeping my heels in line with my hips and shoulders feels right. I'm pretty certain it would help the horse max perform in athletic events - if I was a much better rider! I also ride then with my stirrup on the ball of my foot and TRY to have my heels down. My feet don't stick as far out as they used to, but still are about 45 deg out - anything straighter turns my legs into solid wood, and I'm convinced a relaxed leg is more important than toes ahead.
With an Australian or western saddle, off property, I've moved my feet a bit forward and am experimenting with my stirrup at mid-foot. If it works for campdrafting, cutting, steeplechase & polo, and if Julie Goodnight does it when riding spooky horses (Question answered by Julie--where to put your foot. . . . | Facebook
), then maybe it is something to add to my bag of tricks. If one is more concerned with the bad things that happen when you lose your stirrup than you are about getting free, then it makes sense. I also am a bit skeptical that it REALLY helps get the foot free better than the home position...but I don't have the experience to judge that well.
I spent most of my adult life in the US Air Force. Flying manuals distinguish between procedures and techniques. Procedures are things you MUST do, or risk death or serious damage to the jet. Techniques are things that may work well for you in some situations, but not others. They are options that you can use or not use as you gain experience.
The riding I did as a teen seemed to be all procedure, no technique. And that may be best, when starting. Returning to riding after a 25 year break, and doing it daily instead of a dozen times each year, I'm trying to learn WHY we do things, so I can tell what things are procedures and which ones are techniques - and I thank all the posters on this thread for giving me a lot to think about.