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.
...."a purely peer-to-peer version of electronic cash"
I'm not certain how it is you are attempting to twist these words but surely it means every user wishing to participate/transact in a truly peer-to-peer manner needs to run a full node.
There is no way around it.
What is "the bitcoin payment network" anyway if not a collection of its users?
"...would allow online payments to be sent directly from one party to another..."
you are assuming that each party is a peer
satoshi said himself that at scale, not everyone would run a full node, that users would run lightweight wallets.
by "bitcoin payment network" I mean the people doing the full node thing.