72 Hour free transit facility for US Citizens: Beijing and Hong Kong

Greetings,
This summer my mom and I (US passport holders) want to fly from Incheon, South Korea to Beijing, stay for one night and fly to Hong Kong to stay for 2 nights, then fly back to Korea. I read the information above and think that our travel plans follow the guidelines for traveling to 3 countries on the 72 hour free transit.
I am basing this understanding off the following: "In addition, Hong Kong, Macau and Taiwan are regarded as a third region in terms of transit issues. If a flight passes USA – Beijing – Hong Kong/Macau/Taiwan, passengers are able to enjoy."
According to the information read, it seems as though we must fly to Hong Kong.
I understand that we are to tell the SK airline at check-in about our travel plans and that we must request permission to leave the airport in Beijing, because we will be touring the Great Wall.
I need to make sure I understand this policy correctly before booking our flights.
Can we use the visa for these travel plans?