Kinda wierd that it works running and not KOEO. Could be a sticking pump as unlikely as it sounds, and when its running the extra voltage gets it going.
First thing to do i guess is see pump relay is getting its signal to supply the 12 volts in KOEO. Id start at the engine bay fuel pump relay because its easy to get to, check the relays low side obviously, i forget offhand if the
ECU connects to earth or gives 12v on pins 85/86, it probably grounds, check the wiring diagrams.
If the low side of the relay is fine then you might want to check for 12v at the pump plug KOEO
Whether its KOEO or running, the fuel pump relay trigger circuit should be getting its signal from the
ECU. I doubt somebody wouldve run power to it differently as they'd have to get quite tricky to signal it to only run when engine is running... I suppose its always possible someone mcgyvered it if the relay trigger signal from the
ECU was playing up, just seems unlikely to me... if someone had done this it just seems more likely to me pump would still turn on when ignition is on, ie KOEO... it just wouldnt stop after a few seconds.