I guess I can swap the cables around on the solenoid and try that. I just didn't want to do that if it would cause more damage.
You mean more damage than using a Ford solenoid instead of a Honda?
I can't really say for sure, but if it were me, I'd at least use a motorcycle solenoid, but that's just me.
ADDED: I put me thinking cap on, and thought of what could be wrong. Tell me, which wire glows red?
The only reason a wire is going to glow red is because it's carrying a large amount of current, much more than the system is designed to carry. So, why is it carrying too much current, other than the obvious that it's hooked up wrong. A 12v solenoid might work on another 12v system, or it might not.
The reason is that the resistance of the coil might be much different. Lets say the resistance of the Honda solenoid coil is 100 ohms, just for giggles. On a 12v system, that coil will draw 12/100 or .12 amps.
Now, lets say the ford coil is 10 ohms. 12/10 is 1.2 amps. A factor of 10 more current. And if it's 1 ohm, you're pulling 12 amps. That might be way too much.
1.2 amps is still a small amount of current in the big picture and wouldn't cause any wire to glow [unless it was a very small wire], so there must be something else wrong and I would do what 2five0 said and check the wiring, but I'd still want to know the difference between the solenoids before I just arbitrarily put a different one on there. Again, that's just me.
ADDED again: This is all about the low current side of the circuit, not the starter side of the circuit.