Why is a password called the trivial case of an authenticator, and what weakness follows from that?

With a password the authenticator is the secret itself: there is no activation value, no fresh input and no real function, so the output sent to the verifier is the secret, and every login hands the secret over.

In the general schema the secret stays inside the authenticator and only a computed output leaves it. A password collapses that structure: output = secret. Everything else follows:

• Replay: anyone who sees the output once (a keylogger, a camera, a phishing page, a compromised server) can log in again with exactly the same value.
• No per-login freshness: since nothing new goes in, the output is identical at every login.
• The verifier sees the secret: even an honest server receives the password in clear (inside TLS) at each login.

Hashing the password on the client does not change this. The hash simply becomes the new password-equivalent, and an attacker who captures it can replay it (pass-the-hash).

Compare a smartcard: its secret, the private key, never leaves the chip, and its output is a signature over a random challenge. Capturing one output is worthless, because the next login uses a different challenge.

Go deeper:

• Challenge-response authentication (Wikipedia) — how a fresh challenge stops replay, in contrast to sending the password itself.