Well, we have one that looks like that in our camper. It works at least two ways.
If there is city water pressure, press down on the handle and water flows. If you want a stream without holding it, there was a long "L" bracket threaded into the counter next to it. The bracket would rotate onto the handle to keep it pressed down. We always boondock, so I unscrewed and stored the hold-down. I'm sure an accumulator would work just the same on a faucet you press as it would a faucet you turn.
When all you have is a tank, you push and pull on the handle, and "row" it back and forth to bring up spurts of water up from the tank. Each cycle gives a good spurt, but it is serious work to fill a pot. My neighbors in the '60s had a two-way that had an "umbrella handle" like older car emergency brakes. You pulled the handle out, and pumped it up and down like a bicycle pump. It was tilted 30° to 45° toward you to make pumping easier. I guess you rotated it to call for city water. It was a lot easier to pump with, but probably fell out of favor due to space issues under the sinks.
The "third" way the website advertises it works is as an "electric pump activator." I can only guess what this means. There may be a switch to enable pump power when you press down into the city position, so you don't have to turn on the pump control. Since there is a two-way version, I expect that the three-way adds the switch contacts. That would be good, as I would probably turn off the pump when sleeping or away, so a rupturing pipe or hose doesn't let the pressure sensor cause the pump to empty the tank onto the floor. Having the shut-off bypassed when you want a quick drink may be worth the extra $5 or $6 extra cost.