This spring I have replaced my lift pump (was still running ok) with another Cummins approved (10 psi) lift pump (I made a little adapter kit, PM me if interested). It runs smoother with this than with the stock filter-mounted pump.
Anyway, I can just barely hear this pump when turning on the key, and I have noticed that usually it runs for about 3-4 seconds, but sometimes it does not run at all. Every time it refuses to run, it starts right away. I have come to the conclusion that the ECM knows if you have fuel pressure, which is hard to believe, given that the rail sensor has a +- 500 psi precision.
My theory is that fuel drains from the rail thru the lift pump (which has a check valve) and thru the injectors, either into the engine (not so good) or into the fuel tank via return line.
When more fuel drains out than usual, let's say something is not perfectly closed, it takes longer to bleed the air out of the lines, and it takes a couple of seconds longer to start.
The solution is to always wait until the lift pump stops, which is about 3-4 seconds, before cranking. You will also hear the injectors clicking, as they are bleeding air bubbles back into the fuel tank. Then it should start in 2-4 turns of the starter.
If you wait too long after the lift pump has stopped, fuel will leak out again, because the injectors are still bleeding the fuel lines back to the fuel tank. Maybe this is what they are reflashing to correct.
When you change fuel filters, bump the starter, which lets the lift pump run for 30 seconds or so, then crank it.
Sometimes, when the engine is hot, there may be other causes, such as battery sensors not matching the inlet air temperature, which confuses the ECM for a couple of seconds, or you could even have vapor lock, like a gasser, if the engine is real hot, like DMork may have had. (or there may be something else wrong with his truck)