[search] allow explicitly disabling web search (#9249)

moving `web_search` rollout serverside, so need a way to explicitly
disable search + signal eligibility from the client.

- Add `x‑oai‑web‑search‑eligible` header that signifies whether the
request can have web search.
- Only attach the `web_search` tool when the resolved `WebSearchMode` is
`Live` or `Cached`.

codex-rs/core/src/lib.rs

@@ -111,6 +111,7 @@ mod user_shell_command;
 pub mod util;
 
 pub use apply_patch::CODEX_APPLY_PATCH_ARG1;
+pub use client::WEB_SEARCH_ELIGIBLE_HEADER;
 pub use command_safety::is_dangerous_command;
 pub use command_safety::is_safe_command;
 pub use exec_policy::ExecPolicyError;