Since we are talking about a UK legal requirement (the 90 day rule), neither ICAO Annex 1 nor JAR-FCL is relevant as neither of them sets down UK law.
The ANO 2009 is the only relevant document and Article 50(1) states:
Subject to the exceptions set out in articles 51 to 60, a person must not act as a
member of the flight crew of an aircraft registered in the United Kingdom without
holding an appropriate licence granted or rendered valid under this Order.
The relevant exemption in this case is Article 53(1), which states:
A person may act as pilot of an aircraft of which the flight crew required to be carried
by or under this Order is not more than one pilot for the purpose of becoming qualified
for the grant or renewal of a pilot's licence or the inclusion or variation of any rating in
a pilot's licence within the United Kingdom, the Channel Islands and the Isle of Man,
without being the holder of an appropriate licence granted or rendered valid under this
Order
A 'trial lesson' is a part of the PPL course and may be counted towards the dual flight instruction required for the issue of that licence. Mr J Public, who is (in law) in the aircraft for the purpose of becoming qualified for the grant of a pilot's licence, can act as a pilot of the aircraft (i.e. a member of the crew).
Consequently, the answer to the question "Please tell me what qualifies Mr J Public to be a member of the flight crew?" is "Articles 50 and 53 of the ANO 2009".