I agree that the rotation of Earth has not an effect with what you see.
The longer comment in AskPhysics is interesting, (but I believe the candle has no effect at all).
As you say correctly, the wheel should be magnetized axially.
But maybe there is a little amount of magnetization in the horizontal sense too, in the wheel.
Maybe, this could happen if the magnets are not perfectly aligned with the wheel, so the magnetic field in the wheel would be a bit asymmetrical.
The axis of the magnetic field would not pass through the center of the wheel but somewhere near it, and, that side of the wheel would
acquire a polarization relatively to the opposite side of the wheel, then the wheel could react to the magnetic field of Earth.
Another possibility is that it is not the wheel that it is reacting to the magnetic field of Earth, but the magnet of the wheel that reacts to the magnet above, to which it is suspended. For this to happen, there must be some misalignment between the magnetic fields of the two magnets, (also because of the little magnets added above, as suggested in that comment ?), and the axis of the magnetic field of the magnet of the wheel should be at least a bit tilted from the vertical, otherwise the magnet above couldn't apply any spinning force to it.
But, since you have been able to spin the wheel with a magnet near it, more probably it is the wheel that it is magnetized, horizontally.
With that magnet you could also see what is the side of the wheel that is attracted from it, and if, that side, would point consistently at the north/south, after having removed all the other magnets around.
If the magnetic field of the wheel is not stable, but changes orientation, this could be due to the magnet of the wheel not being fixed steadily, maybe it can spin/tilt a bit relatively to the wheel, and, when this happens, the magnetic field in the wheel changes orientation.