The 04 pumps have a check ball (relief valve) in the side of them. It is just a press fit, nothing more that holds it in. The 04 truck I have worked on several times had all the same symptoms you describe, and it turned out that this "check ball" popped out. Truck ran great one day, and the next day it was dead. New HPOP from International and the truck was back on the road.
On edit, here is a picture of where the check ball is located on the 04 6.0 HPOP....
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99 F350 XLT 4x4, Crew Cab, 19.5s, Cummins 12V powered, with a few mods, SBC DD clutch
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