fix: handle all web_search actions and in progress invocations (#9960)

### Summary
- Parse all `web_search` tool actions (`search`, `find_in_page`,
`open_page`).
- Previously we only parsed + displayed `search`, which made the TUI
appear to pause when the other actions were being used.
- Show in progress `web_search` calls as `Searching the web`
  - Previously we only showed completed tool calls

<img width="308" height="149" alt="image"
src="https://github.com/user-attachments/assets/90a4e8ff-b06a-48ff-a282-b57b31121845"
/>

### Tests
Added + updated tests, tested locally

### Follow ups
Update VSCode extension to display these as well

codex-rs/exec/tests/event_processor_with_json_output.rs

@@ -20,6 +20,7 @@ use codex_core::protocol::PatchApplyEndEvent;
 use codex_core::protocol::SandboxPolicy;
 use codex_core::protocol::SessionConfiguredEvent;
 use codex_core::protocol::WarningEvent;
+use codex_core::protocol::WebSearchBeginEvent;
 use codex_core::protocol::WebSearchEndEvent;
 use codex_exec::event_processor_with_jsonl_output::EventProcessorWithJsonOutput;
 use codex_exec::exec_events::AgentMessageItem;
@@ -54,6 +55,7 @@ use codex_exec::exec_events::TurnStartedEvent;
 use codex_exec::exec_events::Usage;
 use codex_exec::exec_events::WebSearchItem;
 use codex_protocol::ThreadId;
+use codex_protocol::models::WebSearchAction;
 use codex_protocol::plan_tool::PlanItemArg;
 use codex_protocol::plan_tool::StepStatus;
 use codex_protocol::plan_tool::UpdatePlanArgs;
@@ -124,11 +126,15 @@ fn task_started_produces_turn_started_event() {
 fn web_search_end_emits_item_completed() {
     let mut ep = EventProcessorWithJsonOutput::new(None);
     let query = "rust async await".to_string();
+    let action = WebSearchAction::Search {
+        query: Some(query.clone()),
+    };
     let out = ep.collect_thread_events(&event(
         "w1",
         EventMsg::WebSearchEnd(WebSearchEndEvent {
             call_id: "call-123".to_string(),
             query: query.clone(),
+            action: action.clone(),
         }),
     ));
 
@@ -137,12 +143,82 @@ fn web_search_end_emits_item_completed() {
         vec![ThreadEvent::ItemCompleted(ItemCompletedEvent {
             item: ThreadItem {
                 id: "item_0".to_string(),
-                details: ThreadItemDetails::WebSearch(WebSearchItem { query }),
+                details: ThreadItemDetails::WebSearch(WebSearchItem {
+                    id: "call-123".to_string(),
+                    query,
+                    action,
+                }),
             },
         })]
     );
 }
 
+#[test]
+fn web_search_begin_emits_item_started() {
+    let mut ep = EventProcessorWithJsonOutput::new(None);
+    let out = ep.collect_thread_events(&event(
+        "w0",
+        EventMsg::WebSearchBegin(WebSearchBeginEvent {
+            call_id: "call-0".to_string(),
+        }),
+    ));
+
+    assert_eq!(out.len(), 1);
+    let ThreadEvent::ItemStarted(ItemStartedEvent { item }) = &out[0] else {
+        panic!("expected ItemStarted");
+    };
+    assert!(item.id.starts_with("item_"));
+    assert_eq!(
+        item.details,
+        ThreadItemDetails::WebSearch(WebSearchItem {
+            id: "call-0".to_string(),
+            query: String::new(),
+            action: WebSearchAction::Other,
+        })
+    );
+}
+
+#[test]
+fn web_search_begin_then_end_reuses_item_id() {
+    let mut ep = EventProcessorWithJsonOutput::new(None);
+    let begin = ep.collect_thread_events(&event(
+        "w0",
+        EventMsg::WebSearchBegin(WebSearchBeginEvent {
+            call_id: "call-1".to_string(),
+        }),
+    ));
+    let ThreadEvent::ItemStarted(ItemStartedEvent { item: started_item }) = &begin[0] else {
+        panic!("expected ItemStarted");
+    };
+    let action = WebSearchAction::Search {
+        query: Some("rust async await".to_string()),
+    };
+    let end = ep.collect_thread_events(&event(
+        "w1",
+        EventMsg::WebSearchEnd(WebSearchEndEvent {
+            call_id: "call-1".to_string(),
+            query: "rust async await".to_string(),
+            action: action.clone(),
+        }),
+    ));
+    let ThreadEvent::ItemCompleted(ItemCompletedEvent {
+        item: completed_item,
+    }) = &end[0]
+    else {
+        panic!("expected ItemCompleted");
+    };
+
+    assert_eq!(completed_item.id, started_item.id);
+    assert_eq!(
+        completed_item.details,
+        ThreadItemDetails::WebSearch(WebSearchItem {
+            id: "call-1".to_string(),
+            query: "rust async await".to_string(),
+            action,
+        })
+    );
+}
+
 #[test]
 fn plan_update_emits_todo_list_started_updated_and_completed() {
     let mut ep = EventProcessorWithJsonOutput::new(None);