There's a sort of subtle difference that plays out in how effective the turning is. In countersteering, you don't move the center of mass of the motorcycle-human system; the wheels pull out from under the bike and it naturally turns into the capsizing. Body steering actually moves the center of mass of the system out of the bike's vertical axis. (That is, you're no longer in plane with it.)

Because motorcycles are so massive (at least mine is), body steering is much less effective, but it does help if you want to lean the bike a bit further. That's sort of where the advice to "push down with your legs comes from", since it shifts the bike down as you push yourself up - voila, bike wheels make a deeper u shape, capsizes a bit more, and turns harder. It's a trim knob for turning, kind of how the rear brakes add stability while the front do most of the work.

Like a lot of the points in the MSF course, it's a useful detail to internalize so that when something inevitably dumb/awful happens on the road a poorly trained snap decision doesn't lead to panic (ohgodohgodohgodthebikeisntswervingfastenoug-), but in everyday life you won't be thinking about it directly.