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/config.rs

@@ -506,7 +506,6 @@ pub struct ProjectConfig {
 
 #[derive(Deserialize, Debug, Clone, Default)]
 pub struct ToolsToml {
-    // Renamed from `web_search_request`; keep alias for backwards compatibility.
     #[serde(default, alias = "web_search_request")]
     pub web_search: Option<bool>,
 
@@ -672,7 +671,7 @@ impl Config {
             })?
             .clone();
 
-        let shell_environment_policy = cfg.shell_environment_policy.clone().into();
+        let shell_environment_policy = cfg.shell_environment_policy.into();
 
         let resolved_cwd = {
             use std::env;
@@ -693,7 +692,7 @@ impl Config {
             }
         };
 
-        let history = cfg.history.clone().unwrap_or_default();
+        let history = cfg.history.unwrap_or_default();
 
         let tools_web_search_request = override_tools_web_search_request
             .or(cfg.tools.as_ref().and_then(|t| t.web_search))
@@ -775,7 +774,7 @@ impl Config {
             codex_home,
             history,
             file_opener: cfg.file_opener.unwrap_or(UriBasedFileOpener::VsCode),
-            tui: cfg.tui.clone().unwrap_or_default(),
+            tui: cfg.tui.unwrap_or_default(),
             codex_linux_sandbox_exe,
 
             hide_agent_reasoning: cfg.hide_agent_reasoning.unwrap_or(false),
@@ -794,7 +793,7 @@ impl Config {
             model_verbosity: config_profile.model_verbosity.or(cfg.model_verbosity),
             chatgpt_base_url: config_profile
                 .chatgpt_base_url
-                .or(cfg.chatgpt_base_url.clone())
+                .or(cfg.chatgpt_base_url)
                 .unwrap_or("https://chatgpt.com/backend-api/".to_string()),
 
             experimental_resume,