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/app-server-protocol/src/protocol/v2.rs

@@ -2621,6 +2621,7 @@ mod tests {
     use codex_protocol::items::TurnItem;
     use codex_protocol::items::UserMessageItem;
     use codex_protocol::items::WebSearchItem;
+    use codex_protocol::models::WebSearchAction;
     use codex_protocol::protocol::NetworkAccess as CoreNetworkAccess;
     use codex_protocol::user_input::UserInput as CoreUserInput;
     use pretty_assertions::assert_eq;
@@ -2728,6 +2729,9 @@ mod tests {
         let search_item = TurnItem::WebSearch(WebSearchItem {
             id: "search-1".to_string(),
             query: "docs".to_string(),
+            action: WebSearchAction::Search {
+                query: Some("docs".to_string()),
+            },
         });
 
         assert_eq!(