Your amount of forwardness while posting or in half seat should be dictated by the saddle you use, pretty much. All saddles have a balance point that is created by the position of the stirrup bars and how the saddle rides on the back of the hrose and such. You can experiment with how far upright or backward you can stand in the stirrups without falling backward. Just in front of that will be the balance point. As your legs and core get stronger, you can ride really close to the balance point. If you are new to this kind of riding or lack the physical strength of the leg, you will have to compensate by leaning further forward.
You are lucky to be able to try all these different things on good, ol' Gidget. She's a gem!
I used to have a trainer that would make me do rising trot (on a lead line) with a crop locked between my elbows and behind my back. This forced my most natural position to occur. I found it to afford me the most natural half seat I ever had.
Gidget is great. She does everything I ask,no joke. She may royally suck but she still tries and that's why I love her.
I talked to my trainer and told her that I noticed I was beginning to lean forward as I am learning to do half seat. She said this is normal but I need to make sure that I'm not leaning over like I'm in half seat or whatever. I worked on it today at the stable on a POA =D His name is Spot and well I think I like my horse a bit taller than 13.2-13.3 hh,lol. He defiently looks a lot bigger but he is a lot of fun and stocky.
I also asked my trainer if I can jump Gidget to see what her opinion was on it and she said that she doesn't see why I couldn't as along as she knows what she is doing so today I got into my half seat and cantered towards a 1 ft. Jump and we did a clean jump with no bumping on the mouth or losing stirrups. I also learned where to place my hands now so I've been working on it every night.
I will get a video and see what you guys think...I just have to remember to get one :P