Can someone ELI5 if the video with the magnet levitating is supposed prove something? Is this visual result only possible with superconductors? If so, why?
All superconductors are strongly (perfectly, actually) diamagnetic, and its a classic cool demonstation of their properties. However, not all strongly diamagnetic things are superconductors. In fact, diagmagnetism is present in all materials, but it is usually swamped by other magnetic effects (ferromagnetism and paramagnetism).
Earnshaw's theorem describes why a free-floating (paramagnetic) magnet cannot levitate over another such magnet. There must be at least one stable axis which is not from a magnet. See https://en.wikipedia.org/wiki/Earnshaw%27s_theorem
> The magnet will induce a current in the superconductor
That's a different sort of levitation. See https://en.wikipedia.org/wiki/Electrodynamic_suspension . We know it's not the same because you can see a magnet floating above a superconductor even though there's no motion.
The relevant reason is https://en.wikipedia.org/wiki/Superdiamagnetism .
] Superconducting magnetic levitation is due to superdiamagnetism, which repels a permanent magnet which approaches the superconductor, and flux pinning, which prevents the magnet floating away.
] Superdiamagnetism is a feature of superconductivity.
> In both of these magnetic repulsion videos, keep in mind that magnetic repulsion and/or levitation are not, by themselves, probative of superconductivity: a diamagnetic material such as pyrolytic graphite [16], can be made to levitate [19] in a magnetic field without being superconductive.
See https://en.wikipedia.org/wiki/Magnetic_levitation#Diamagneti... .