Remove the deprecated Claude subscription login path and restore a green Rust workspace

ROADMAP #37 was still open even though several earlier backlog items were
already closed. This change removes the local login/logout surface, stops
startup auth resolution from treating saved OAuth credentials as a supported
path, and updates diagnostics/help to point users at ANTHROPIC_API_KEY or
ANTHROPIC_AUTH_TOKEN only.

While proving the change with the user-requested workspace gates, clippy
surfaced additional pre-existing warning failures across the Rust workspace.
Those were cleaned up in-place so the required `cargo fmt`, `cargo clippy
--workspace --all-targets -- -D warnings`, and `cargo test --workspace`
sequence now passes end to end.

Constraint: User explicitly required full-workspace fmt/clippy/test before commit/push
Constraint: Existing dirty leader worktree had to be stashed before attempted OMX team worktree launch
Rejected: Keep login/logout but hide them from help | left unsupported auth flow and saved OAuth fallback intact
Rejected: Stop after ROADMAP #37 targeted tests | did not satisfy required full-workspace verification gate
Confidence: medium
Scope-risk: moderate
Reversibility: clean
Directive: Do not reintroduce saved OAuth as a silent Anthropic startup fallback without an explicit supported auth policy
Tested: cargo fmt --all --check; cargo clippy --workspace --all-targets -- -D warnings; cargo test --workspace
Not-tested: Remote push effects beyond origin/main update

rust/crates/api/src/error.rs

@@ -24,7 +24,7 @@ pub enum ApiError {
         env_vars: &'static [&'static str],
         /// Optional, runtime-computed hint appended to the error Display
         /// output. Populated when the provider resolver can infer what the
-        /// user probably intended (e.g. an OpenAI key is set but Anthropic
+        /// user probably intended (e.g. an `OpenAI` key is set but Anthropic
         /// was selected because no Anthropic credentials exist).
         hint: Option<String>,
     },