Leviton GFCIs constantly trip on shaded pole motors
So do all shaded pole motors trip on any GFCI?


No. There is nothing in the design of a shaded-pole motor or any other type of AC motor that makes them particularly vulnerable to tripping a GFCI. There are electronic speed controllers (ESC) that can create a vulnerability, but that is not relevant to your situation.


  Does anyone know what kind of rotor/stator is the following?


You have already determined that it is a shaded-pole motor. That is correct and the YouTube video that you linked is mostly correct The rotor is usually described as steel with aluminum bars rather than aluminum with steel bars. That particular style is often called a C-frame shaded-pole motor.


  Since there is no way for the current to be lost to ground. What could be other mechanisms?


I would suspect a problem with the GFCI or wiring. Check the Waterpik in other GFCI outlets and other loads in the outlet in question.


  Could it be inductor at one wire taking time to go to zero so it creates temporary imbalance? Is this an accepted explanation?


I don't think so. There may be some problem with the wiring that causes a GFCI problem when something is switched on. That would be worth researching.


  What kinds of particular motor types can trip GFCI akin to the above?


There are none except for the ESC mentioned above.