1995 Jeep Cherokee car starts, but not the right way

OK, car wouldn't start one day. Hadn't driven it in a week, so I thought it was the battery. Swapped out the battery with my other jeep, still wouldn't start. Doesn't turn over, but the buzzer works with the key in the ignition and the door open. Had a mechanic come look at it and he said it is the starter. It is a motorcraft starter so I know the solenoid is a separate part. Could it be the solenoid and not the starter? It will start by running a wire directly from the starter and touching it to the battery with the key in the on position. I didn't do this, somebody my daughter knows figured this out. Thanks.