Welcome to the forum, definitely not breaking any rules, this is a cool project!
I have a few thoughts that may help in debugging the pump:
You want to run it at the first resonance of the disk a so the higher order modes will not be the disk deforming up and down like in the diagram, it will instead be a much more interesting mode shape. The second eigenmode will probably be a symmetric and so there won’t be a change in volume in the chamber.
From the looks of the diagram the pumping mechanism seems to assume that the central chamber is full and that the vibrations are moving the water (air is compressible so ruins the efficiency). I cannot tell from your video but it looks like there is not much water in the system. (Of course filling it to the top with 130V going to the piezo is going to take some thinking about risk, some sort of dealing membrane may be worth considering)
I am not sure it will work pumping from the current containers, as unless the the output container is filled above the hole, air will seep into the system. What if you made the input chamber a bit larger, and gave it a lid. Then remove the output chamber so it is just a spout, and made it bit longer than in the current design. The aim would be to for the system to have the liquid in the input chamber not leaking out of the end by surface tension on the end of the spout, so it would probably require a fairly small hole. You would also need some clever way to tilt the system to purge all air from put pump container (hence the lid in the input chamber).
I assume the pumping force is pretty weak so you need to find another way to rely on surface tension. I Would not be surprised if it takes quite a lot of optimisation of the shapes and sizes of the components to get a noticeable force. The group that did this work seem to do a lot of simulations to get something with a pumping speed of a few uL/min.
Good luck, let us know how it goes.