the ecm commands the prime for a few seconds after key on by powering up the fuel pump relay on passenger fender , after the prime times out the relay deactivates and pump goes off . after keying off for 10 seconds and then turning key on will start the prime cycle again.
as you crank the starter if the ecm sees a sufficient tach signal it commands the pump relay again
it has nothing to do with pressure
now there is a secondary pump supply at the oil sending switch (its also the idiot lamp switch) that will provide power to the pump once oil pressure goes above 4psi (turns off idiot lamp above 4psi oil pressure)
its possible that connector which is rubber is crudded up and shorting the power feed at that switch to the pump supply feed and the pump stays running or the sender switch is stuck (not common)
unplug it and see if pump still runs
not knowing whos been monkeying with your car or what its history is its possible someone miswired a hotwire relay or plugged something into the fuel prime connector behind the alternator (by the ac lines ) or bypassed the fuel relay to have the pump run with key on thats keeping the pump powered
