I'm amazed at how complicated everyone seems to think this is!
If the company or organisation or person conducting the flights is charging any other entity for the service, it is Air Transport, requires an AOC, and the pilot needs a CPL. If not, then it's private (Part 91), and the minimum pilot qualification is a PPL.
As for hiring the aircraft, the company hiring a machine from its owner is akin to renting a car - they're just paying for an asset, not a service. Charter would be paying for the service - equivalent to the taxi or limo.
I deal with CASA staff ever week, and I've neverhad any of their FOIs or Regservices staff unclear on such fundamentals. Its certainly much clearer than the old CAR 206 classifications.
Having said that, insurance is the bigger elephant in the room. AOC operators are required to hold Carriers Liability Insurance and have a measure of protection under the law. Private operators are not, which means in the event of an accident, the pilot may be just as good a target as the company / directors, so think carefully as PIC about what insurance you want for carrying random people around in the back.