NONE OF THE ANIMATIONS SHOWN ABOVE ARE CARTOONS--all of the field configurations and magnet dynamics are calculated quantitatively.
In the case where an experimenter determines the motion of the magnet as he or she pushes it into the coil or pulls it out of the coil, we have solved a single differential equation for the current in the coil given the motion of the magnet. In the case where the magnet is falling through the conducting ring, the current induced in the ring by the motion of the magnet in turn affects the motion of the magnet, so that we must solve a coupled set of equations. The equations can be reduced to a set of three first order ordinary coupled differential equations for the current in the ring, the position of the magnet, and the speed of the magnet. For details of the mathematics, see the PDF document at this link.