[codex] Enable standalone web search in code mode (#26719)

## What

- Consume plaintext `output` from standalone search while retaining
optional `encrypted_output` parsing.
- Expose `web.run` to code mode and return search output to nested
JavaScript calls.
- Cover direct and code-mode standalone search paths with integration
tests.

## Why

`/v1/alpha/search` now returns plaintext output, which code mode needs
to consume standalone search results.

## Test plan

- `just test -p codex-api`
- `just test -p codex-web-search-extension`
- `just test -p codex-core code_mode_can_call_standalone_web_search`
- `just test -p codex-app-server
standalone_web_search_round_trips_output`

codex-rs/app-server/tests/suite/v2/web_search.rs

@@ -41,7 +41,7 @@ const DEFAULT_READ_TIMEOUT: Duration = Duration::from_secs(60);
 const DEFAULT_READ_TIMEOUT: Duration = Duration::from_secs(10);
 
 #[tokio::test]
-async fn standalone_web_search_round_trips_encrypted_output() -> Result<()> {
+async fn standalone_web_search_round_trips_output() -> Result<()> {
     let call_id = "web-run-1";
     let server = responses::start_mock_server().await;
     mount_search_response(&server).await;
@@ -170,8 +170,8 @@ async fn standalone_web_search_round_trips_encrypted_output() -> Result<()> {
             "type": "function_call_output",
             "call_id": call_id,
             "output": [{
-                "type": "encrypted_content",
-                "encrypted_content": "ciphertext",
+                "type": "input_text",
+                "text": "Search result",
             }],
         })
     );
@@ -259,6 +259,7 @@ async fn mount_search_response(server: &MockServer) {
         .and(path("/api/codex/alpha/search"))
         .respond_with(ResponseTemplate::new(200).set_body_json(json!({
             "encrypted_output": "ciphertext",
+            "output": "Search result",
         })))
         .expect(1)
         .mount(server)