Traveling form USA to South Korea transfering in two cities of China.

I am from Chile and my flight itinerary is Houston (USA)-Beijing-Hangzhou-Jeju (South Korea). I will stay for less than 9 hours in China but I have two stops (PEK and HGH). Do I need a visa? Air China claims that I need but now that I read this web page (numeral 2 -For direct transit) I am confused.