Gotcha...I think I have a better idea in my head of what you're trying to tell us.
If I'm following you correctly, it sounds like you have 12v from battery going to your starter. You turn the key, but nothing happens, no 12v from solenoid to starter. You jumped your solenoid on the starter from constant 12v cable to the other side of the solenoid and your starter cranked.
Given the above, this means one of two things...your solenoid on the starter is bad and needs to be replaced or the 12v signal from the starter switch isn't making it to the solenoid when you turn the key. You need to check voltage at the start wire on the solenoid when the key is turned to 'start.' If you get 12v at the start wire, then you have a bad solenoid. If you don't get 12v at the start wire, then you have an issue up-river...probably at the ignition switch as stated in your other thread; it's not uncommon. Or it could be your starter relay.
When I say 'start wire,' don't mean the wire from the solenoid to the starter. I mean the wire from the ignition switch to the solenoid on your starter.
If the start wire isn't providing 12v when you turn the key, it won't close the circuit in the solenoid.
Wiring diagrams are worth a thousand words. If you can trace each wire and draw it out for us, we can tell you what's going on with a little more certainty.
On to solving your mystery piece. I re-read your original post, and I may have followed you incorrectly. If all it has is 12v from battery to a post, another wire from that same post to the starer, and a ground, then it looks like the PO made a modification to accommodate the starter he/she installed that has a solenoid on it. If the hot wires on your mystery part share the same post, then the mystery part is doing absolutely nothing other than just being there instead of one longer wire from the battery to the starter.
Let me know if I'm understanding this correctly.