-Does anyone know of or has anyone experienced this problem and actually solved it?

Yes. Exactly the same symptoms occurred to my sister's Cherokee. It was parked for about an hour at the Olive Garden on Wetmore in Tucson. She was meeting for lunch with some ladies from her craft club. On the sidewalk, later, in the hot sun, she paused after the luncheon to discuss some macrame styles with one of the ladies while the rest left. Her conversation continued at the window of this gal's SUV for a few moments until she drove off.

When my sister opened up her Cherokee and let the hot air out, the key did nothing - just a small click and then silence.

Having spent a childhood around me and cars, she knew what to do. She popped the hood and peered in at the motor some until a gentleman approached. He offered to let her use his cellphone, but instead, she turned on her headlights and asked him to turn the key and try to start it himself. When he did, the headlights went out. She wiggled the red cable on the battery, then the black, and the headlights came back on. Then he turned the key and started the Cherokee.

She thanked him profusely and rode home in air conditioned comfort. At home she got out her tool kit and cleaned her battery terminals.