I think I found something.
In the first drawing there is the tip of a top spinning, (the top is very tilted for more clarity), the top spins clockwise and the top moves to the left.
This is the normal situation.
Below instead we have the tip of the Hart top.
The tip spins clockwise, as the tip above. In the meanwhile it vibrates.
Vibration is due to the center of mass of the flywheel and the center of mass of the shell misaligned and spinning around the center of mass of the whole top.
So the vibration of the shell is a tiny and rapid circular motion in the same direction of the flywheel, (counterclockwise).
If the speed and intensity of the vibration are important enough relatively to the spin speed of the shell, it happens that, in the instants when the tip touches the dish, the contact point of the tip will be moving towards the left instead of towards the right, and consequently the tip will be pushed towards the right.
In fact, looking at your second video of the Hart top at 1:30, it can be seen that the top walks
in the opposite direction than it should if we just consider the spin direction of the shell. The vibration reverses the direction of the push of the tip on the dish.