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/exec/src/event_processor_with_human_output.rs

@@ -81,8 +81,8 @@ impl EventProcessorWithHumanOutput {
                     "started".style(self.dimmed)
                 );
             }
-            ThreadItem::WebSearch { query, .. } => {
-                eprintln!("{} {}", "web search:".style(self.bold), query);
+            ThreadItem::WebSearch(item) => {
+                eprintln!("{} {}", "web search:".style(self.bold), item.query);
             }
             ThreadItem::FileChange { .. } => {
                 eprintln!("{}", "apply patch".style(self.bold));
@@ -196,8 +196,8 @@ impl EventProcessorWithHumanOutput {
                     eprintln!("{}", error.message.style(self.red));
                 }
             }
-            ThreadItem::WebSearch { query, .. } => {
-                eprintln!("{} {}", "web search:".style(self.bold), query);
+            ThreadItem::WebSearch(item) => {
+                eprintln!("{} {}", "web search:".style(self.bold), item.query);
             }
             ThreadItem::ContextCompaction { .. } => {
                 eprintln!("{}", "context compacted".style(self.dimmed));

codex-rs/exec/src/event_processor_with_jsonl_output.rs

@@ -293,16 +293,12 @@ impl EventProcessorWithJsonOutput {
                     },
                 }),
             }),
-            ThreadItem::WebSearch {
-                id: raw_id,
-                query,
-                action,
-            } => Some(ExecThreadItem {
+            ThreadItem::WebSearch(item) => Some(ExecThreadItem {
                 id: make_id(),
                 details: ThreadItemDetails::WebSearch(WebSearchItem {
-                    id: raw_id,
-                    query,
-                    action: match action {
+                    id: item.id,
+                    query: item.query,
+                    action: match item.action {
                         Some(action) => serde_json::from_value(
                             serde_json::to_value(action).unwrap_or_else(|_| json!("other")),
                         )

codex-rs/exec/tests/event_processor_with_json_output.rs

@@ -27,6 +27,7 @@ use codex_app_server_protocol::TurnPlanUpdatedNotification;
 use codex_app_server_protocol::TurnStartedNotification;
 use codex_app_server_protocol::TurnStatus;
 use codex_app_server_protocol::WebSearchAction as ApiWebSearchAction;
+use codex_app_server_protocol::WebSearchItem as ApiWebSearchItem;
 use codex_protocol::SessionId;
 use codex_protocol::ThreadId;
 use codex_protocol::models::PermissionProfile;
@@ -362,14 +363,14 @@ fn web_search_completion_preserves_query_and_action() {
 
     let collected = processor.collect_thread_events(ServerNotification::ItemCompleted(
         ItemCompletedNotification {
-            item: ThreadItem::WebSearch {
+            item: ThreadItem::WebSearch(ApiWebSearchItem {
                 id: "search-1".to_string(),
                 query: "rust async await".to_string(),
                 action: Some(ApiWebSearchAction::Search {
                     query: Some("rust async await".to_string()),
                     queries: None,
                 }),
-            },
+            }),
             thread_id: "thread-1".to_string(),
             turn_id: "turn-1".to_string(),
             completed_at_ms: 0,
@@ -403,11 +404,11 @@ fn web_search_start_and_completion_reuse_item_id() {
 
     let started =
         processor.collect_thread_events(ServerNotification::ItemStarted(ItemStartedNotification {
-            item: ThreadItem::WebSearch {
+            item: ThreadItem::WebSearch(ApiWebSearchItem {
                 id: "search-1".to_string(),
                 query: String::new(),
                 action: None,
-            },
+            }),
             thread_id: "thread-1".to_string(),
             turn_id: "turn-1".to_string(),
             started_at_ms: 0,
@@ -415,14 +416,14 @@ fn web_search_start_and_completion_reuse_item_id() {
 
     let completed = processor.collect_thread_events(ServerNotification::ItemCompleted(
         ItemCompletedNotification {
-            item: ThreadItem::WebSearch {
+            item: ThreadItem::WebSearch(ApiWebSearchItem {
                 id: "search-1".to_string(),
                 query: "rust async await".to_string(),
                 action: Some(ApiWebSearchAction::Search {
                     query: Some("rust async await".to_string()),
                     queries: None,
                 }),
-            },
+            }),
             thread_id: "thread-1".to_string(),
             turn_id: "turn-1".to_string(),
             completed_at_ms: 0,