Response to enginboy's Post: Hi, Enginboy:
I think your answer is very close to the truth.
Except my Battery was very good.
I took out the starter and dissembled it. The solenoid was worn out. The contact was much thinner. It was barely make contact. That was why the new symptom was not starting, the solenoid just clicked.
I tried to push the solenoid to engage position to see if it can stuck there, it couldn't. So the welding explaination was a very good one. Before the solenoid contact went bad completely, it was barely making contact. This could be a very good recipe for temporary welding of the contact. It kept the starter engaged (cranking) even after the key is removed. Because the "welding" was not strong, after 10 or 20 seconds when key is removed, it seperated under the force of restoring spring.
I have replaced the starter for about 2 weeks now and I drove it everyday. The car feels as good as new when starting. No problem at all.
Thank you for your reply. I still have one dollar in my account, I can not find the button to pay it to you. Please let me know how I can get it done.