starter relay
THAT is one heck of a mess.
Modified wiring is very difficult to help troubleshoot because "we have no idea" what you have.
This still gets back to the basics and the "path."
The ignition switch outputs IGN1 (only with key in run) and IGN2 (ballast bypass) (only with key in start). The IGN2 is the ignition power source in "start" and the IGN1 "run" is the power source when the key springs back to that position. That path takes both circuits out into the engine bay. IGN2 ONLY goes to the coil side of the ballast, IGN1 branches off various points and also feeds one end of the ballast. MAKE SURE both those work as they should and go from there