That's a really beautiful video.
The effect is a little more complicated than simple exclusion of the magnetic fields from the superconducting regions. As you noticed, that simple effect would just leave the superconductor sort of bouncing around on the magnetic field, as if it were supported by a low-friction mattress. This puck, however, can get stuck in particular positions, as if it were actually impaled on the mattress springs.
That, in fact, is not a bad model for what's going on. That's a Type II superconductor, in which magnetic field can enter as little tubes, called vortices, which locally disrupt the superconductivity. These vortices get stuck on various imperfections in the material, so it really acts as if it were penetrated by some springy wires. With enough applied force, the vortices can be yanked away from the pinning sites, but then they'll settle down and get stuck again when the force is small.
I think that standard description fits what you see in the video.
(published on 10/20/2011)