Following up on #2371 post commit feedback (#2852)

- Introduce websearch end to complement the begin 
- Moves the logic of adding the sebsearch tool to
create_tools_json_for_responses_api
- Making it the client responsibility to toggle the tool on or off 
- Other misc in #2371 post commit feedback
- Show the query:

<img width="1392" height="151" alt="image"
src="https://github.com/user-attachments/assets/8457f1a6-f851-44cf-bcca-0d4fe460ce89"
/>

codex-rs/core/src/conversation_history.rs

@@ -72,7 +72,7 @@ fn is_api_message(message: &ResponseItem) -> bool {
         | ResponseItem::CustomToolCallOutput { .. }
         | ResponseItem::LocalShellCall { .. }
         | ResponseItem::Reasoning { .. } => true,
-        ResponseItem::Other => false,
+        ResponseItem::WebSearchCall { .. } | ResponseItem::Other => false,
     }
 }