No there are some errors in your statements
When on the ground with the engine master sw’s both not off or both not on the PTU will only run when the parking brake is on and the NWS is in not in the towing position!
The PTU will only run when the P Brake is set to OFF not ON, i.e. when your moving, your second statement seems to get this right.
Additionally with no engines running (both master sw OFF), if the yellow electrical pump is switched on, the green system will pressurise via the PTU, which is why ground crew must be advised before switching it on and the PTU can be set to OFF first to avoid this.
Precisely
The PTU is inhibited whenever the Y Pump pb is OFF, whenever a cargo door is operated until 40seconds after the door stops operation and during first engine start.
Be careful on this one, look at the diagram, notice the shape of the box that contains the lines of the inhibit conditions (Y Pump is off, cargo door is operated). Its an AND gate, not an OR gate, meaning that the inhibit condition is when the Cargo door is operated AND the Y Pump pb is off, not either or, both have to occur at the same time.