Ghost Start Up

1998 Toyota Camry LE, Mileage <120000, Automatic, 4 cylinder, 2 Wheel Drive
My car broke down a couple days ago, here is the scenario. I go out to my car like any other day and try to start it up, but it doesn't start. I then cranked it again and about half way through starting up everything just dies, i.E. No cabin light, radio, prestart, nada. So my friend and I look under the hood and the battery cable terminals are horrendously corroded, so we clean those and replace both Positive (+) and Negative (-) terminals. We reconnect the Negative (-) terminal to the battery, but when we go to connect the Positive (+) terminal, (with no keys in the ignition) to the post the starter turns over like its getting ready to start. When the ignition key is turned to the on position and we reconnect the Positive (+) battery terminal and the car actually starts and then dies after about 2 seconds. We called out a mechanic the other day to diagnose the problem and he stated the ignition switch had shorted out and was stuck in the "on" position. I wanted to get a second diagnoses from two "pros". Thanks -D