I've got to agree with Old-Biscuit, when the faucet is closed and pressure on the discharge side of the pump reaches a preset level a diaphragm presses against a little rod that opens a switch that supplies power to the pump motor. If the check valve is leaking or a fixture is leaking the pressure drops and the switch is closed which in turn turns on the pump motor.
If that pressure switch is staying open for a second or two too long when the faucet is opened, then the pump won't run. I had a very similar situation in the Winnie which was caused by a leak above the pump head and water was getting into the pressure switch causing rust to build up and the little rod to stick and eventually oxidation ruined the switch.
One thing that does puzzle me a bit is, the OP says, " No water comes out of the tap until the pump is turned on". I don't know what he means by turning on the pump. The pump is normally left on and the action of the pressure switch turns the pump on and off. If the OP means that they have to turn the ON/OFF to the off position and then on again, I would bet that it is the contacts of the pressure switch or possibly but very unlikely the ON/OFF switch are bad.
Low current draw switches can and do corrode and become high resistance. When the power is cycled on and then off, there is an instantly high draw which causes the contacts to conduct and will continue to conduct until opened by the pressure in the pump head.
I'd remove the pressure switch disassemble and inspect the switch, if there is corrosion or damage, replace the pressure switch assembly or the entire pump.
