This has piqued my interest and I suspect that the instructors reply was based upon a general understanding - not having read the specifics.
Firstly The Standardised European Rules of the Air (commonly referred to as SERA) took effect across Europe on 4 December 2014 and in the UK superseded most (but not all) of the UK Rules of the Air Regulations 2007.
The old Rule 17(2) states - 2) The commander of an aircraft arriving at or departing from an aerodrome in the United Kingdom shall take all reasonable steps to ensure, upon landing or prior to departure, as the case may be, that the person in charge of the aerodrome or the air traffic control unit or flight information service unit at the aerodrome is given notice of the landing or departure. which is what most people claim sets the requirement for 'booking out'.
The equivalent SERA clause - which does fully replace the old rules of the air states - SERA.4015 Changes to a Flight Plan
(a) Subject to the provisions of SERA.8020 (b) all changes to a flight plan submitted for an IFR flight, or a VFR flight operated as a controlled flight, shall be reported as soon as practicable to the appropriate air traffic services unit. For other VFR flights, significant changes to a flight plan shall be reported as soon as practicable to the appropriate air traffic services unit.
(b) Information submitted prior to departure regarding fuel endurance or total number of persons carried on board, if incorrect at time of departure, constitutes a significant change to the flight plan and as such shall be reported.
So the emphasis in very much on flight plans or controlled flights - not booking out where borders are not crossed which has been the application by many in the UK.