You lost me on the moveable ball of muscle, I'm familiar with the concept of dantien, but not exactly with it being what you described here, I think I'm just not understanding what you mean. If there is a unique ball of muscle, that you can see under the skin, couldn't we just take a quick video of that ball? I really don't think I understand your description.
So, are you saying that the muscles don't contract at all during the leg press? That there is literally no muscle being used to do the work of pressing the leg out?
So I think you said you once had access to Forrest Chang's SJT video. He demonstrated something that looks straight out of the movie alien. So in the movie alien, before the alien pops out of the persons chest, you can see its head moving below the skin. Someone with a developed enough dantien will have a ball of muscle that can move around like that. Its not the upper abs, as just about anyone should be able to pulls those around. If its not that conditioned, it won't be as visible.
If you go on kendo-world, you will find discussion ny some 7th dans (I think the guys account name is chidokan) saying they would place their hands on an 8th dan named Iwata (passed away a few years back), and that you would feel his tanden/dantien move around to drive the sword forward and back for cuts, so tanden driven movement is certainly more than just a broad reference to the hara in modern iaido/kendo, even if in my opinion, few if any of the high level kendo/iaido guys I have seen can actually use it.
So when you are at the gym, you can target certain muscles, the leg press is the same way, where one targets the muscles associated in the area of the dantien. I'm not going to be more specific than that per requests, but dantien specific conditioning is well known (conceptually at least) in chinese martial arts. The results of it should be dantien/tanden driven/initiated power.
I'm of the opinion that aiki-ken style suburi, which looks different than typical japanese swordsmanship type suburi was designed to be utilized for dantien driven movement. Much the same for Ueshiba's garden/farmwork and Sagawa's spear thrusts.
Now I do want to say that I don't think everyone who is studying IS is focused on dantien driven movement, and thus there are different levels of purity. Someone who can access a groundpath has a foot in the door into learning IS and arguably could be considered to have some (low level) understanding of it, but they don't have dantien driven movement.