thats not how the water pump works but okay. It has two 1 way valves. one on the piston and on the base. There has to be a sealed tube going from the pump down into the water. Sucking the air out of the tube will force the water to come up with eatch pump.
First you pump down and that sucks water past the first valve which closes when the down stroke is complete. Then when you life the handle the water will go through the valve on the piston and get stuck. So now when you push back down the water will come out the end. A little bit of water stays on each valve to help keep a better seal to not let air through. It would be extremely hard to create a sealed well so that you could increase the air pressure enough to rise the water how many feet rather than create a vacuum in the pipe which is a much smaller volume
Whoever proof-read this missed some spots. Quite a few places have the words strung together or misspelled. I feel for the MC though, I too make things out of convenience but don't really want to bother with mass production.