Can't do better than DC's last post, mate!
... although I would add that quite a few "unpaid" activities involving pax (the words "gratuitously carried" have remained in the rules) may or may not also be definable as "public transport" depending on whether they satisfy one or more of the exemptions in Part 34.
God help us if the UK tries to paste these arcane PT rules on top of the "new" OPS, currently under consideration. Imho, they are well past their sell-by date.
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Mind you, all this is miles off topic... I thought the term "safety pilot" was coined to massage the ego of the PPL-holding owner who wanted to play at 'engine drivers' in a SP jet.
Cheers,
Sepp.