Everyone is going to tell you something different for these, as everyone is taught different.
Two-point is what you usually see over fences or when galloping. There are two points of contact(both feet), hence the name. You are out of the saddle and bent at your hips.
Three-point, also called full seat, is "normal" riding position. There are three points of contact, both feet and your seat.
Half seat is in between two- and three- point. It is commonly seen between fences in hunters. Your feet are in the stirrups and you are slightly raised out of the seat. For lack of a better term, your crotch is sort of "grazing" the saddle.
Like others have said, once you understand the basics and rhythm of posting, the motion will come from your thighs and not your feet. I think of it as a forward-back motion as opposed to up-down.
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