An aircraft that is flying or operating for the purpose of, or in the course of, the personal transportation of the owner of the aircraft is, for the purposes of the regulations, taken to be a private operation. So if, for example, Pauline Hanson is the owner of an aircraft, she can pay any pilot anything she likes to fly her around in the aircraft for her personal transportation, and it’s a private operation.
But not so if the pilot owns the aircraft instead of Pauline.
(It makes it easier to understand the classification of operations scheme once you realise that the factors which determine the various classifications have nothing to do with objective safety or the acceptance of understood risks.)