US Citizen flying from Seoul, KR to Qingdao, China and returning to Seoul the next day

Well, you can see by yourself that this is not a transit, but a return trip, so you are not eligible for any kind of visa free transit scheme and you need a Chinese visa obtained in advance. However, if you wish, you can somewhat alter your itinerary to look like:

Seoul - Tokyo - Qingdao - Seoul

Immediate points before and after your mainland Chinese destination have to be in two different countries or eligible territories (eligible territories: Hong Kong, Taiwan and Macau).