I had a similar issue with my 02 Silverado 1500 5.3. The truck wouldn't turn over the first time I cranked it. I'd have to turn the key allowing the fuel pump to operate, then turn it back off, then on again like you're doing. It eventually got worse and worse. I replaced the FPR with no change in the problem. I then had my fuel pump and fuel filter replaced. The problem improved by having this done, but was never completely solved. Up until I got rid of it, I would start it like a diesel. Turn the key to on, pause, then crank it. If I turned the key too quickly it would be a hard start. Or if I turned the key to on and left it there for too long, I'd have to turn it back off before cranking it or the truck wouldn't start.
I also eventually had a section of fuel line replaced secondary to corrosion. That didn't improve the problem either.
Sorry I'm not much help, but I wanted to let you know you aren't alone. It always drove me nuts too because I never figured out the problem. I solved it by buying a new truck. Time will tell if that was a good fix!
