I'm not sure what is meant by the stirrup being the center of gravity, so I can't address that. I'm not even sure how it became a point of discussion, since the OP was asking aout hand position...
In post 33, the OP further wrote:
As for leg placement, I've had a MAJOR chair seat for the past two years so considering that, that picture is pretty decent comparing. Anyway, in this picture my legs are too far back right?
That is where the discussion shifted to include leg position.
Mildot is correct about what I'm trying to say. The stirrup is normally very close to the center of gravity of the rider for English riding. The amount of weight the rider has in front of the stirrup should equal the weight behind it if the rider wants to be balanced with weight on the stirrup.
A chair seat describes when your center of gravity is behind the stirrups. If the horse was suddenly removed, you would not land standing - just as you would not if someone suddenly yanked a chair out from under you. But if your horse is likely to make many sudden stops, or many very sharp turns, or if you want to use a rope, then a chair seat helps - which is why many stock saddles, western and Australian, are designed to put you in a chair seat. It is good for working cattle. And many find it more comfortable, since it requires less physical effort from the rider (per Littauer).
But if you want your horse to go fast in a straight line, or to jump, you need to free up his back. For almost any kind of racing or speed events, and for any jumping, a chair seat hurts performance. Thus the forward seat. However, as you shorten the stirrups, you cannot maintain heel-hip alignment AND keep you weight balanced at the stirrup bars (and stirrups).
If your weight is centered on the stirrups, your lower leg can be still and you are balanced. But with your hip and heel aligned, as you fold up, your body is now mostly forward of the stirrups and you must support your weight with the horse's neck - which is worse than supporting it with his back.
Perhaps this is a better racing picture:
A vertical line drawn from the stirrup splits the jockey in two. With just his toe in the stirrup, the length of his foot brings his heel closer to his hip, but there is not a vertical line from his heel to hip.
An advantage, and this brings us back to the original post, is that if your weight is balanced over the stirrup, then you won't use the reins for support. You don't need them to balance, because you ARE balanced.
In the OP's picture, the stirrup leathers are bent at an odd angle. If you estimate where her stirrup bars are, then her weight is a little behind the stirrup bars, which is normal at that pace. If I get my stirrup leathers bent like that, it usually means I am gripping with my knees and not letting my weight fall uninterrupted into my heels - but I don't know if that is true of the OP. I tend to be a bad 'knee-gripper', myself.
My point is that riding is about balance. Hip-heel (or hip-heel-shoulder) discussions are static. Trying to make a hip-heel (or HHS) alignment focuses on the wrong aspect of riding, and may have nothing to do with balance.
And the more the body needs to fold, the less the hip and heel can be aligned with balance. If the stirrup straps hang straight down, and the rider's body is centered on the stirrups, then the rider is well balanced for speed or jumping.
English saddles meant for jumping put the stirrup bars further forward to allow the rider to balance over them at speed or in a jump. A dressage saddle will put it further back, because dressage horses aren't being raced - so the horse's center of gravity is further back. A western saddle often hangs the stirrups further forward, but that is because having weight further back (chair seat) is good when you rope a steer.
Per my signature, having the right balance for the task at hand is what we need to focus on - not alignment of our body parts. The biggest fault of most of the riding books I own is that they focus on rider position and not rider balance. One of the wonderful things about practicing a two point, even in a western saddle, is that you MUST focus on balance and moving with the horse.
For the OP question in post 33, the best answer, IMHO, are questions:
"a) are you in fluid balance and rhythm with your horse or not?
B) does your seat enable you to control your horse efficiently?"
- V.S. Littauer