Does purely peer-to-peer refer to every single user, or just the bitcoin payment network? My reading of that statement is that purely peer to peer refers to the nodes that make up the payment network, not that every end user needs to run a node.
The definition itself is a bit contradictory.
I guess, Satoshi gave us a puzzle rather than a solution.
Not that
every user
should be running a node, but
enough users
should be able to run one.