In order to claim that one is the "owner" of the aircraft then one must have title to the aircraft......it could be for a limited time but it must be legally completed...........not just a hire agreement - eg you do not own a hire car when you hire it.
However, I believe it is perfectly legal to hire a car and then have your chauffeur drive it for you. This does not make it a taxi operation. An imperfect analogy I realise.
From previous debate, FAA law and European practice say, if the punter legitimately sources an aircraft (buys, leases, rents, borrows) and then has his hired help fly the plane this isn't public transport.