I have the EXACT same symptoms as you had. I was told by 'HVAC,' and also my Dodge Diesel friend visiting from Texas, to not start my truck when the 'wait to start' light goes out, but wait another '6 or 7' seconds and then try to start it.
It's worked every time so far, even when it's warm. 'HVAC' says that the 'VP44' doesn't like high(15psi+) pressures when hot, so, if you wait a few more seconds, it might start right up. (If not. it's probably the VP44)
So far, even though I have a brand new pump ready to install, I'll just hold off a few more days or even weeks to see if 'HVAC's ' advice hold's true!
Bob in Sacramento