Something is wrong here. It is impossible for a starter to work if a battery cable is disconnected. I have a suspicion someone else disconnected the smaller wire at the starter, and something got lost in translation.
The multiple symptoms suggest the starter solenoid is sticking. That has been somewhat of a common problem on '87 and newer GM products. The solenoid housing warps a little, then the metal plunger sticks and may not fully release or it may not fully engage. It's job is to pop a gear out to engage the engine, THEN it turns on a large switch to turn on the starter motor. If it sticks and doesn't release, the starter can continue cranking the engine after the ignition switch is released. If it sticks when you try to start the engine, a jump-start can provide just the little more voltage that is needed to get it to engage. You can do the same thing by tapping on the solenoid while a helper is holding the ignition switch in the "crank" position.
The fix is to replace the starter solenoid, but given the cost and the additional time, the better solution is to just replace the entire starter. They always come with a new solenoid.
Be aware that unlike other forums where anyone can chime in and confuse the issue, here you are in a private conversation. Once you added a second comment, we see your question on the list as having already received a reply, so most of us will skip right over it without reading it. Also, it can take many hours to get a reply. We are only here at various times, so don't panic if it takes a while. We'll stay with you for days or weeks until the problem is solved.
Wednesday, April 5th, 2017 AT 4:40 PM
