While what you're saying makes sense, it's not the normal use of the term - in fact, the term 'end to end encryption' was basically coined to differentiate user-to-user encryption (through an intermediary service that can't decrypt the message) from the regular case (user to service encryption) that you're talking about!

It wasn't coined, it was reused. It historically meant things that were encrypted from the client to the server, e.g. SSH, SSL, TLS, etc.

RFC 4949 (Internet Security Glossary, Version 2) from 2007: https://datatracker.ietf.org/doc/html/rfc4949

     $ end-to-end encryption
      (I) Continuous protection of data that flows between two points in
      a network, effected by encrypting data when it leaves its source,
      keeping it encrypted while it passes through any intermediate
      computers (such as routers), and decrypting it only when it
      arrives at the intended final destination. (See: wiretapping.
      Compare: link encryption.)

      Examples: A few are BLACKER, CANEWARE, IPLI, IPsec, PLI, SDNS,
      SILS, SSH, SSL, TLS.

      Tutorial: When two points are separated by multiple communication
      links that are connected by one or more intermediate relays, end-
      to-end encryption enables the source and destination systems to
      protect their communications without depending on the intermediate
      systems to provide the protection.