If we cool a superconductor S below it's critical temperature while a magnet is resting on top of it, an electric current I1 will develop on it's surface to expel de magnetic field. Now let us suppose we first cool S then approach the magnet. My reasoning tells me that this time there will be an induced current too, because of the motion of the magnet in the neigbhorhood of S. So S will not only have to counter the field (diamagnetism) but it will also have to counter it's increase (induction, Lenz's law). So it seems to me that this time the current I2 will be greater than I1. Is this reasoning correct ?
In short, yes. The superconductor resists the change in both directions, this can be seen when people have mobius strips with superconductors, or simply when flipping a unit of magnet+sc together