Faraday’s law doing what? -  

Magnets are magnetized radially. Full picture is way more complicated than it could have been introduced and explained in a minute ( there might follow up a explainer video). Firstly there exist currents both as shown in the video and also azimuthal direction aswell ( here created by stray axial magnetic field). As you said interesting demonstrations can be made making cuts in copper ring but also varying height of it, and thickness.

@@MHDTechnologyLaboratory dev/nul is correct. The way the ring drops in the center of the large magnet shows that it is magnetized axially (vertical N/S on top bottom). If it was dominantly radial, there would be induced eddys all along the drop. Near the ends of the large mag, the field is spreading, thus 'cutting' the ring. Fleming's Right and Left Hand Rules are the fundamental principles at play. It is very common for people to show those circular eddy currents when the dominant net is actually azimuthal (around the ring's circumference). This forms in the rings vertical poles. It is a 'shorted turn'. .. In essence, the ring forms a magnetic mirror which guarantees fields that oppose the movement, thus demonstrating how the most fundamental, first principles of Fleming laws build up to make the higher level phenomenon we call Lenz's Law.