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/core/src/codex.rs

@@ -3439,10 +3439,8 @@ async fn try_run_sampling_request(
             }
             ResponseEvent::OutputItemAdded(item) => {
                 if let Some(turn_item) = handle_non_tool_response_item(&item).await {
-                    let tracked_item = turn_item.clone();
                     sess.emit_turn_item_started(&turn_context, &turn_item).await;
-
-                    active_item = Some(tracked_item);
+                    active_item = Some(turn_item);
                 }
             }
             ResponseEvent::ServerReasoningIncluded(included) => {