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/ext/items/src/tests.rs

@@ -3,6 +3,8 @@ use serde_json::json;
 
 use super::ExtensionItem;
 use super::image_generation::ImageGenerationItem;
+use super::web_search::WebSearchAction;
+use super::web_search::WebSearchItem;
 
 fn completed_image_generation_item() -> ExtensionItem {
     ExtensionItem::ImageGeneration(ImageGenerationItem {
@@ -35,6 +37,37 @@ fn image_generation_item_preserves_stable_wire_shape() {
     );
 }
 
+#[test]
+fn web_search_item_preserves_stable_wire_shape() {
+    let item = ExtensionItem::WebSearch(WebSearchItem {
+        id: "search-1".to_string(),
+        query: "docs".to_string(),
+        action: Some(WebSearchAction::Search {
+            query: Some("docs".to_string()),
+            queries: None,
+        }),
+    });
+    let value = serde_json::to_value(&item).expect("serialize extension item");
+
+    assert_eq!(
+        value,
+        json!({
+            "kind": "web.search",
+            "id": "search-1",
+            "query": "docs",
+            "action": {
+                "type": "search",
+                "query": "docs",
+                "queries": null,
+            },
+        })
+    );
+    assert_eq!(
+        serde_json::from_value::<ExtensionItem>(value).expect("deserialize extension item"),
+        item
+    );
+}
+
 #[test]
 fn unknown_extension_kind_is_rejected() {
     let value = json!({