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/exec/src/event_processor_with_human_output.rs

@@ -25,6 +25,7 @@ use codex_core::protocol::TaskCompleteEvent;
 use codex_core::protocol::TurnAbortReason;
 use codex_core::protocol::TurnDiffEvent;
 use codex_core::protocol::WebSearchBeginEvent;
+use codex_core::protocol::WebSearchEndEvent;
 use owo_colors::OwoColorize;
 use owo_colors::Style;
 use shlex::try_join;
@@ -362,8 +363,9 @@ impl EventProcessor for EventProcessorWithHumanOutput {
                     }
                 }
             }
-            EventMsg::WebSearchBegin(WebSearchBeginEvent { call_id: _, query }) => {
-                ts_println!(self, "🌐 {query}");
+            EventMsg::WebSearchBegin(WebSearchBeginEvent { call_id: _ }) => {}
+            EventMsg::WebSearchEnd(WebSearchEndEvent { call_id: _, query }) => {
+                ts_println!(self, "🌐 Searched: {query}");
             }
             EventMsg::PatchApplyBegin(PatchApplyBeginEvent {
                 call_id,