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/rollout.rs

@@ -135,7 +135,7 @@ impl RolloutRecorder {
                 | ResponseItem::CustomToolCall { .. }
                 | ResponseItem::CustomToolCallOutput { .. }
                 | ResponseItem::Reasoning { .. } => filtered.push(item.clone()),
-                ResponseItem::Other => {
+                ResponseItem::WebSearchCall { .. } | ResponseItem::Other => {
                     // These should never be serialized.
                     continue;
                 }
@@ -199,7 +199,7 @@ impl RolloutRecorder {
                     | ResponseItem::CustomToolCall { .. }
                     | ResponseItem::CustomToolCallOutput { .. }
                     | ResponseItem::Reasoning { .. } => items.push(item),
-                    ResponseItem::Other => {}
+                    ResponseItem::WebSearchCall { .. } | ResponseItem::Other => {}
                 },
                 Err(e) => {
                     warn!("failed to parse item: {v:?}, error: {e}");
@@ -326,7 +326,7 @@ async fn rollout_writer(
                         | ResponseItem::Reasoning { .. } => {
                             writer.write_line(&item).await?;
                         }
-                        ResponseItem::Other => {}
+                        ResponseItem::WebSearchCall { .. } | ResponseItem::Other => {}
                     }
                 }
             }