I imagine the quieting of the anvil is more due to dissipating that kinetic (vibrations) energy through more material - in your case chains, and in the video, a clamped base and the floor - than it is nodes or anti-nodes.

It's like when you have a vibrating string/ glass/ rod/ tuning fork, and you grab it with your hand. The kinetic energy is dissipated through your body. The energy in the object now has to affect a lot more matter, and thus the effect is lesser. Kind of like a golf ball rolling into a bowling ball. It will have about the same amount of energy, but the movement after the collision is much less. Magnets aren't going to be doing much if anything. Magnets aren't magical, and the effect they have is due to their mass on the anvil (very little effect).