Your video didn't stay on the gauge long enough. Normally the pump would cut in about 40 psi and turn off about 60 psi, again, depending on how it was set. Observe the gauge and see what the cut in and cut out pressures are with a faucet running.
TO DO IT RIGHT: Turn the breaker off to the well pump. If you have ball valves going to different parts of the house, shut them down . That will keep water from backing into the tank. Connect a hose to the spigot on the tank and drain the tank. Once it is drained, take a tire pressure gauge and see what the tank pressure is. It should be 2 lbs lower than the cut in pressure, so if the switch is set at 40 psi, you should have 38 lbs of air in the tank. If low, charge it with a compressor a little at a time. Turn the ball valves back on and turn the breaker back on. Note the cut in and cut out pressures on the gauge. They should be in line with what is printed on the label in the cap of the switch.,