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/tests/suite/client.rs

@@ -1211,9 +1211,9 @@ async fn azure_responses_request_includes_store_and_reasoning_ids() {
     prompt.input.push(ResponseItem::WebSearchCall {
         id: Some("web-search-id".into()),
         status: Some("completed".into()),
-        action: WebSearchAction::Search {
+        action: Some(WebSearchAction::Search {
             query: Some("weather".into()),
-        },
+        }),
     });
     prompt.input.push(ResponseItem::FunctionCall {
         id: Some("function-id".into()),