I've not heard of a actual definition of a private flight ( CAA or EASA).
The operator is not defined from ownership.The CAA has never had the authority to demand advance notice of a non-scheduled flight into the UK. A flight plan is required for international flights of course but they are not routinely forwarded to the CAA and do not require permission for VFR.
The home Office/Border Force have the power to prevent the entry if it has good reason but that wouldn't include defining the category of the flight. The Dept. of Transport/CAA do that but have no immigration or customs powers. The CAA cannot prevent a flight unless they believe the aircraft is unworthy, the pilot is not qualified or is unfit and the flight is believed to being operated illegally. They must not obstruct the free movement of flight operating properly in accordance with ICAO.