[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/tests/suite/prompt_caching.rs

@@ -88,7 +88,7 @@ async fn prompt_tools_are_consistent_across_requests() -> anyhow::Result<()> {
             config.user_instructions = Some("be consistent and helpful".to_string());
             config.model = Some("gpt-5.1-codex-max".to_string());
             // Keep tool expectations stable when the default web_search mode changes.
-            config.web_search_mode = WebSearchMode::Cached;
+            config.web_search_mode = Some(WebSearchMode::Cached);
         })
         .build(&server)
         .await?;