Does Bose's 'nuance' simply apply to the need for an AOC?
If I remember correctly a flight can be a private flight provided that the only payment made is for the services of a pilot. Since, in this Thread, the aircraft is being rented neither of these flights could be considered 'private'.
There is also the question of an organisation wanting to know, probably to check out, any pilot who will fly/command their aircraft.
In the second scenario it is not possible to 'Cost Share' with the PIC because the flight has not, actually, cost the pilot anything.
The simplest way of achieving this would be for the pilot to rent the aircraft and then to share the cost of this (and any other flight costs) with any passengers they fly with - Now, under EASA, there are no percentages mandated so the pilot only has to pay 1pence/1cent with their passengers paying the rest.