Does precession (of a top for example) occur only whenever the angular velocity vector is not parallel to a principal axis? I have yet found any clear definition of precession and when it is defined to occur. If so, why does that imply that whenever the angular velocity vector is parallel to the angular momentum vector, there will not be any precession?
From the point of view of the physics taking place: When the spin axis and the torque axis coincide then any work that the torque is doing will go into changing the rate of spin, instead of inducing precession.
In order to be in a position to induce precession the axis of the torque must be non-aligned with the spin axis.
Whatever definition is used, it must be compatible with the above.