I haven't worked on one of these. But, since it's MPI, I'd think that turning the Key from OFF to RUN, should cause the High Pressure (Injection) Fuel Pump to run a few seconds to prime gasoline right up to your Injectors so you can enjoy that quick start. Which, by the way, I miss! We just replaced a TBI MerC engine with a Carbureted one...
OK, back to you. I'm pretty sure the Priming feature I described, is based on Time, not Pressure. In other words it runs for say 02 seconds under the assumption that gasoline is there to pump, and the injectors will get primed in that time.
What I don't know, is how the gasoline is made available to that high pressure pump. Our old TBI MerC had an engine mounted mechanical pump that filled a Vapor Separator Tank where the High Pressure Electric Injection Pump was located. IF IF IF you have another pump, feeding that Injection Pump, and it isn't doing its job, you'd have to wait for it to "catch up" before the Injector Pump could do its job. MerC Master Tech replaced our expensive Injection Pump when in fact the engine mounted mechanical pump had gone weak. Pump (whole VST assembly, actually) I should put up For Sale.
See if you can pull up a Parts Breakdown or a Service Manual.
EDIT: I also "assume" it has a Pressure Regulator to feed Injectors at a steady pressure that is less than what it receives from the Injection Pump. If that Regulator is leaking, then the Fuel Rail may not pressure up all the injectors for an easy start, within the couple seconds the ECM runs the Pump before Engine Starts.