When 72 hours counting starts? Does it start from arrival time on airline ticket?

We arrive in Beijing on Jan 15th at 11:20am and depart on Jan 18th at 16:00pm. Do we qualify for transit visa? Airline is saying no. I emailed Chinese consulate in San Francisco and they are considering official arrival time? That makes it 77.2 hours.

If we calculate next day 00:00 then it is less than 72 hours.

Also can we go outside Beijing for day sightseeing?