It most likely will depend upon how the airplane is registered. If it's a privately registered plane, being flown by the owner, or lent to someone else, it's private use. If it's a commercially registered plane, that commercial designation invokes the hours/calendar limitation on the engine, no matter by whom or how the plane is being flown.
The 'Transport' and 'Private' designations no longer apply in the UK nor the EU and haven't for many years. A UK aeroplane is issued with a Certificate of Airworthiness whatever the purpose. The maintenance required though is dependent on the aircraft use. If the aircraft is used for gain (this doesn't only mean payments) then the aircraft must be maintained by a licenced engineer and a 'Release to Service' issued. If used solely for private use then many items of maintenance can be done by the owner and recorded in the log book/s, a Release to Service is not required for this.
With regards to engine hours a Release to Service can be issued up to the plus 20% (if allowed) on condition. I'm not sure about the 12 year calendar limit, this has been an issue for debate for some time now. It raised its head some years ago regarding imports. The CAA would not issue a C of A for engines of 12 years old. I'm not sure whether this has been resolved for ongoing use.
Perhaps someone knows what the Australian regulations are for certain.