That being said even if the operation is a Passenger Transport Operation it still may not be an Air Transport Operation if it's not for hire or reward, just to complicate things.
Precisely one of my points.
That's precisely why I am trying to wangle out of CASA where the 'reward' comes from in a flight for which the PIC bears the entirety of the direct costs. CASA's opinion seems to be that a flight which satisfies all of the elements of the definition of 'cost-sharing', other than the one about who bears what parts of the direct costs because, in CASA's opinion, it does not cover the circumstance where the PIC bears the entirety of the direct costs, is not a 'cost-sharing' flight and is, therefore, a 'passenger transport operation'.
Even if all of that is accepted as correct (based on the plain words of the definition, I don't accept that the flight is not 'cost-sharing') it does not follow that the flight is for 'hire or reward'. If the flight is not 'for hire or reward', it is not an 'air transport operation' even if it's a 'passenger transport operation'.
So, I can 'up' the maximum seat configuration of the aircraft I'm flying to e.g. 9 and everyone will say, correctly, that the flight can't meet the definition of 'cost-sharing' flight. Great. So it's a 'passenger carrying operation'.
But if there's no 'hire or reward' in circumstances in which I bear the entirety of the direct costs of the flight and nobody pays me or gives me anything in connection with the flight, it's not an 'air transport operation'. (This, in turn, is why I'm so concerned about CASA's power to start 'recapturing' operations into Part 119 by simply putting a sentence in a MOS.)