There are so many variations of whats out there now I would need to know a lot more about your set up to be more helpful. Example: I had a Ford in here 2 weeks ago with similar symptoms, the original eng. had a starter solenoid mounted on the back of the engine and a "normal" Ford starter without the solenoid, someone had replaced the starter with a newer style permanent magnet starter with built in solenoid, had one solenoid pulling in the second solenoid, the first one had worn contacts inside & was dropping voltage to the second one & would only start when the battery was on a charger. Eliminated one of the solenoids & was good to go. "I" is ignition & is often used as a bypass circuit to feed the ign. coil during cranking, but may not be used on aftermarket electronic ignitions.