Two types of precession?

It occured to me recently that there are two forms of precession.

If you hold a spinning bicycle wheel and force the axis of rotation to tilt on its side, this will induce a torque tending to spin the wheel about a vertical axis. This is in accord with the fact that the change in angular momentum is in a vertical direction.

On the other hand, in the experiment where a spinning bicycle wheel is supported on one end of the axis by a rope and undergoes precession, the direction of the induced rotation is not the same as the direction of the induced torque (which changes over time).

I have always understood the second experiment as an example of precession. Would one put the first example into this same category?