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/codex-api/src/sse/responses.rs

@@ -291,7 +291,7 @@ pub fn process_responses_event(
                 if let Ok(item) = serde_json::from_value::<ResponseItem>(item_val) {
                     return Ok(Some(ResponseEvent::OutputItemAdded(item)));
                 }
-                debug!("failed to parse ResponseItem from output_item.done");
+                debug!("failed to parse ResponseItem from output_item.added");
             }
         }
         "response.reasoning_summary_part.added" => {