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/chat_completions.rs

@@ -129,7 +129,9 @@ pub(crate) async fn stream_chat_completions(
                     "content": output,
                 }));
             }
-            ResponseItem::Reasoning { .. } | ResponseItem::Other => {
+            ResponseItem::Reasoning { .. }
+            | ResponseItem::WebSearchCall { .. }
+            | ResponseItem::Other => {
                 // Omit these items from the conversation history.
                 continue;
             }
@@ -623,11 +625,8 @@ where
                 Poll::Ready(Some(Ok(ResponseEvent::ReasoningSummaryPartAdded))) => {
                     continue;
                 }
-                Poll::Ready(Some(Ok(ResponseEvent::WebSearchCallBegin { .. }))) => {
-                    return Poll::Ready(Some(Ok(ResponseEvent::WebSearchCallBegin {
-                        call_id: String::new(),
-                        query: None,
-                    })));
+                Poll::Ready(Some(Ok(ResponseEvent::WebSearchCallBegin { call_id }))) => {
+                    return Poll::Ready(Some(Ok(ResponseEvent::WebSearchCallBegin { call_id })));
                 }
             }
         }