Author here.

Lamport simply calls his protocol "Paxos" to refer to both the single‑decree and multi‑decree versions. This is also the case in his other works, e.g., "Fast Paxos" and "Generalized Paxos." The term "Multi‑Paxos" is a later community/industry shorthand for the repeated or optimized use of single‑decree Paxos.

“Paxos” is a term that can mean many different things, so it’s better not to get too attached to any one meaning especially in different contexts.

Multi Paxos is commonly used (especially in industry) as short hand for multi decree Paxos (in contrast to single decree Paxos), but “Paxos” most often refers to the family of protocols, all of which are typically implemented with a leader. It is confusing of course because single decree Paxos is used to implement EPaxos (and its derivatives).

It’s worth noting also that Lamport is (supposedly) on the record as having intended “Paxos” to refer to the protocol incorporating the leader optimisation.

In practice, almost every implementation of Paxos uses multi-paxos. Even the "Paxos Made Simple" paper notes:

> In normal operation, a single server is elected to be the leader, which acts as the distinguished proposer (the only one that tries to issue proposals) in all instances of the consensus algorithm.

because otherwise you don't have a mechanism for ordering; the more basic Paxos protocol only discusses how to arrive at consensus for a single proposal, not how to assign numbers to them in a reasonable way that preserves ordering.