If we're discussing bar magnets, the second video is correct: The bar magnet should experience a torque causing it to align with the magnetic field (and then overshoot if friction is low). Think of a compass needle.
If we're discussing electrons however, which we are in the case of Stern-Gerlach, there is another factor in play. Electrons have an intrinsic angular momentum called spin. So we can imagine electrons as tiny bar magnets glued to the inside of a spinning flywheel, so that the north south axis of the magnet is the same as the axis along which the flywheel rotates. When we put such a contraption in a vertical magnetic field, classical physics predicts that the magnetic field acting on the bar magnet should produce a torque, which in turn causes the spinning flywheel to precess. The result it that it rotates about the vertical axis, and does not align with the magnetic field.
Quantum mechanics means that this picture of electrons does not produce correct predictions, however. The key differences should be explained in the video, I believe.