Yes, so that messages can reach a user even if the receiving and sending users are never online at the same time (a server could be another user/peer(s) that isn't part of the communication, so it could possibly still be called p2p?)
That is incorrect. You've just defined P2P such that no commonly accepted P2P application meets it. At a minimum every P2P app you've likely ever heard of or used, baring maybe one exception, has at a minimum used third parties for orchestration.
so if other users (not part of the conversation) are used as temporary servers so that messages get delivered even if the sender or receiver are never connected at the same time, what would it be called?
AFAIK there isn't a modern word to describe the common architecture that you describe.
About a decade ago, the phrase "hub-and-spokes network" was thrown around to describe an architecture with a central server that acted as the mediator. The term "p2p" became popular in this context. (
Napster, bittorrent, tox are all "p2p" because they don't connect to single server. Skype is not "p2p" because all packets go through the skype servers.