Ok then. That tells us a lot. If you removed one or both of the small wires from the solenoid, and it still turned over, that tells you that it was a problem with the solenoid, and not a problem with the pre-start relay. Because, removing the small wires from the solenoid prevents any power from getting to the solenoid from the pre-start relay in the first place, and accidentally energizing it when it is not supposed to. There's something wrong internally with the solenoid. I'd feel comfortable replacing the solenoid at this point.
But if you want to be doubly sure, remove the red wire that comes from the battery from the solenoid stud, such that there is nothing at all on that stud. Use your meter, set it to continuity testing, and take a reading across the two large studs on the solenoid. One stud has nothing on it at all, and the other has the starter motor wire. And that's ok. If it reads close to 0 ohms, you have a internal short on the solenoid. Go ahead and replace it. After that, cut it open and see what happened if you want.