core: migrate standalone web search to extension-owned turn items (#31525)

## Description

This PR migrates standalone web search onto the extension-owned
turn-item path introduced in #31283.

Standalone web search now emits `ExtensionItem::WebSearch` through
generic `TurnItem::Extension`, while app-server still exposes the
existing typed `ThreadItem::WebSearch` JSON shape. Hosted Responses API
web search stays on core-owned `TurnItem::WebSearch`.

## What changed

- Added `web_search::WebSearchItem` and `WebSearchAction` to
`codex-extension-items` under the stable `web.search` kind.
- Collapsed `ExtensionTurnItem` to generic `{ item, legacy_events }` now
that no typed extension special cases remain.
- Kept the existing `WebSearchBegin` / `WebSearchEnd` compatibility
events and canonical-first ordering.
- Updated app-server projection/history and generated TypeScript; the
app-server JSON schema is unchanged.

codex-rs/analytics/src/analytics_client_tests.rs

@@ -140,6 +140,7 @@ use codex_app_server_protocol::TurnStatus as AppServerTurnStatus;
 use codex_app_server_protocol::TurnSteerParams;
 use codex_app_server_protocol::TurnSteerResponse;
 use codex_app_server_protocol::UserInput;
+use codex_app_server_protocol::WebSearchItem;
 use codex_login::default_client::DEFAULT_ORIGINATOR;
 use codex_login::default_client::originator;
 use codex_plugin::AppConnectorId;
@@ -4401,11 +4402,11 @@ async fn turn_event_counts_completed_tool_items() {
             agent_thread_id: "thread-child".to_string(),
             agent_path: "/root/child".to_string(),
         },
-        ThreadItem::WebSearch {
+        ThreadItem::WebSearch(WebSearchItem {
             id: "web-1".to_string(),
             query: "codex".to_string(),
             action: None,
-        },
+        }),
         ThreadItem::ImageGeneration(ImageGenerationItem {
             id: "image-1".to_string(),
             status: "completed".to_string(),