Don't forget that there are multiple contacts being made in that switch. It could just be that holding it half way is allowing the fuel pump to still be energized in the "start" mode, but the starter motor circuit had already been released. The fuel pump circuit may be a "make before break" circuit, where it makes contact with the running circuit before it breaks contact with the starting circuit.

Make sure they check the output on that auxiliary terminal on the alt when you have it checked. That terminal (marked "L" I believe) is the one that controls the fuel pump and main relay.