I think you have exactly the right term already. The Institute of Electrical Engineers use "gravity energy storage" and wikipedia has a Gravity Battery page.
The main issue is that pumping and storing huge amounts of water is a mature technology - people have been doing that for ages. And since gravity isn't very energy dense storing decent amounts ...
The potential energy of anything falling from a height is given by U = mgh, where:
U is the potential energy in joules
m is the mass in kilograms
h is the height in meters
Now, 1000 liters of water has a mass 1000 kg and the value of g is 9.8 m/s^2.
The amount of energy 1000 L of water would have after falling 20 m is,
U = 1000(20)9.8 = 196,000 J
The industry term is "pumped storage" and it's used all over the place at a large scale.
Some systems, like Oroville Dam, reverse their turbine-generators to be "motor-pumps", pumping water from the bottom of the dam to the top*.
The Niagara Falls installation sits well above the Falls, and has separate pumps: they pump from the high ...