Black wire with the red stripe only gets power while cranking the starter. Notice this is (or should be!) connected to the same end of the ballast resistor that the coil is connected to, so this bypasses the ballast during starting to give a bit more oomph to the spark. Weird that simply moving that wire would cause anything to work, so maybe there's a short or break in that wire somewhere?

Black wire with the white stripe, connected to the other end of the ballast resistor, gets power whenever the key is on, so that's the one you'd want to use for triggering the relay. Having the relay ground thru that temp switch would be correct to run the fan only when that switch closes with the key on; my apologies I hadn't considered that before.