I think I understand how the battery where they drop a big weight down a mountain works; how do these work? Or how does it compare in effectiveness as I assume it's probably the same principle?

They need to pair this with Tidal power

https://en.m.wikipedia.org/wiki/Tidal_power

They're going to...pull a vacuum in a concrete sphere deep underwater. And then use the force of water being sucked back in to turn a turbine.

...sure.

Interesting concept, but not very scalable. It's basically a reversed dam - when it's full, there's 0m head of water. Then with excess energy, you lower the level inside, storing the energy in the water outside. E.g -2m head. Water then flows in to equalise head, and doing so, regenerates electricity. Adding depth to supercharge pressure differentials is a good idea, although I wonder how they limit the flow rate, or otherwise prevent cavitation shocks each cycle.

Could be useful as a private industrial battery, but a dam would still be better on an infrastructural level.

Why not submerge a tank with a hole at the bottom and blow air in the tank via a hose to store energy?