Many boilers still have diaphragms to prove pump pressure.
Those that use electronic pressure sensors tend to to be unreliable due to blockages.
You put those words in the wrong order.
The pump proving switch prevents the gas valve being operated until after there is pressure from the pump.
I don't think I did get them wrong. Turn on the hot water tap causes a water pressure differential, which lifts the diaphragm in the water pressure differential valve. This activates the pump (I observed this). The second diaphragm, gets lifted because the pump is running, activating another microswitch and energises the gas valve. That is what I wrote in that order.
You stated there was only one diaphragm. The MI clearly show two and looking at the valve it looks like one is under a cover on top of the 3 way valve.
The 3 way valve has a copper weep tube going to it. I don't know what that is for.