^^^ What Megan said.
Remember that the regulator sees its responsibility first to regulate on behalf of the "public". Thereafter, and to a lesser degree, for "private" individuals. So, if a service is "publicly" available (meaning that the prospective passenger has no "private" relationship to the pilot), it'll be more heavily regulated as a "commercial" flight. The regulator can hardly account for every permutation of flights, so they draw a line. A PPL is taught which side of that line to stay on (pilot cannot be paid), CPL's are taught additional requirements associated with commercial operation expectations.
If a person is boarding a flight, does not know the pilot personally, and is paying for the flight, it's a commercial flight. The pilot is required to be at least a CPL. Other OC requirements may apply too - but it is not a "Private" flight.