[cart around]

That doesn't make sense.

One terminal is connected directly to ground. That means that when checking voltage between it, and the negative battery terminal, it should be zero all the time.

The other small terminal is connected to the positive side of the battery, but the voltage has to go through the key switch, and the pedal switch first. It should only have voltage when those two switches are making contact.

You need to figure out where that constant voltage is coming from.
Here's the part that doesn't make sense. If your solenoid was good, and there really was a constant voltage at that small terminal, your starter would be turning all of the time.

I will also ask just to clarify. You did put the meter leads on the small terminals, and not the big ones, right?
One more thing. Your meter leads were backwards. That's why you got negative numbers.