Question about 72 hour VISA free visit Beijing

We will fly from Amsterdam to Zürich and from Zürich to Beijing.
In Beijing we will arrive at 5 am. We stay there for 2 nights and leave the 3rd day at 23.45 Pm from Beijing AirPort. We will fly to Sydney with another airline company and we will fly the same day also from Sydney to Auckland.

When I am understand the procedure well, we do not need a visum. We need to go to the 72 hours lane with our passport, we fill out a form, we have to show oud tickets from Beijing to Sydney and Sydney to Auckland and after that the hotel will inform the police. It is not permitted to leave the city only to visit the grand wall is permitted. Please can you inform me if I understood it well. Thank you in advance.