By generator, do you mean a
gasoline powered generator?
If yes, then what you say is possible, but there may be more efficient ways of accomplishing the same task. Every time you change the form of energy, you lose some efficiency. So let's just list the conversions in your proposed example, and let's assume you get to keep 80% of the input energy of each step, due to conversion inefficiency:
Gasoline to rotary mechanical energy 100% x 0.80 (that's 80% efficiency) = 80% left
Rotary mechanical energy to electricity via generator 80% x 0.80 = 64% left
Electricity to mechanical rotary energy (an electric pump of some kind) 64% x 0.80 = 51.2%
Rotary pump energy to moving water to a higher elevation 51.2% x 0.8 = 40.9%
So, the best case scenario with 4 conversions is the loss of 60% of the original input energy.
In real life, the losses will be much worse. Most small gasoline engines are
lucky if they turn 25% of the chemical energy of gasoline into useful rotary torque/energy. That's a 75% loss in the first step. Plugging that in drops it to an 88% loss of the original energy in the gasoline.
One improvement would be to drive the pump directly from the gas engine. This cuts out two of the four conversions.
If you could link to one of the videos you refer to, I could make a better estimate. There are also many many different kinds of pumps you could use, which will dramatically affect how well the system as a whole works.
If you could give me an idea of how many liters or gallons of water per day you would need to move, that would help also.