Have I missed something?
Yes. You would be correct were it not for para 3(c) below. From Art 130:
2 Subject to the provisions of this article, an aircraft in flight shall for the purposes of this Order be deemed to fly for the purposes of public transport:
...
(c) for the purposes of Part III of this Order (other than articles 14(2) and 15(2) thereof), if valuable consideration is given or promised for the primary purpose of conferring on a particular person the right to fly the aircraft on that flight...
3(c) Notwithstanding the giving or promising of valuable consideration specified in subparagraph (2)(c) in respect of the flight or the purpose of the flight it shall:
(i) subject to sub-paragraph (ii), for all purposes other than Part III of this Order; and
(ii) for the purposes of articles 14(2) and 15(2) of this Order;
be deemed to be a private flight.