No, you don't. The EASA definition of a cross-country flight is: "a flight between a point of departure and a point of arrival following a pre-planned route, using standard navigation procedures." There is no mention of landing anywhere, nor that the point of departure and point of arrival have to be different.
Well, the following quote is also from CAP804:
FCL.210.A PPL(A) — Experience requirements and crediting
(a) Applicants for a PPL(A) shall have completed at least 45 hours of flight instruction in aeroplanes, 5 of which may have been completed in an FSTD, including at least:
(1) 25 hours of dual flight instruction; and
(2) 10 hours of supervised solo flight time, including at least 5 hours of solo cross-country flight time with at least 1 cross-country flight of at least 270 km (150 NM), during which full stop landings at 2 aerodromes different from the aerodrome of departure shall be made.
(b) Specific requirements for applicants holding an LAPL(A). Applicants for a PPL (A) holding an LAPL(A) shall have completed at least 15 hours of flight time on aeroplanes after the issue of the LAPL (A), of which at least 10 shall be flight instruction completed in a training course at an ATO. This training course shall include at least 4 hours of supervised solo flight time, including at least 2 hours of solo cross-country flight time with at least 1 cross-country flight of at least 270 km (150 NM), during which full stop landings at 2 aerodromes different from the aerodrome of departure shall be made.
My interpretation of cross country emphasizes landing at a distant airport though.
WP