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/protocol/src/models.rs

@@ -95,6 +95,22 @@ pub enum ResponseItem {
         call_id: String,
         output: String,
     },
+    // Emitted by the Responses API when the agent triggers a web search.
+    // Example payload (from SSE `response.output_item.done`):
+    // {
+    //   "id":"ws_...",
+    //   "type":"web_search_call",
+    //   "status":"completed",
+    //   "action": {"type":"search","query":"weather: San Francisco, CA"}
+    // }
+    WebSearchCall {
+        #[serde(default, skip_serializing_if = "Option::is_none")]
+        id: Option<String>,
+        #[serde(default, skip_serializing_if = "Option::is_none")]
+        status: Option<String>,
+        action: WebSearchAction,
+    },
+
     #[serde(other)]
     Other,
 }
@@ -162,6 +178,16 @@ pub struct LocalShellExecAction {
     pub user: Option<String>,
 }
 
+#[derive(Debug, Clone, Serialize, Deserialize, PartialEq)]
+#[serde(tag = "type", rename_all = "snake_case")]
+pub enum WebSearchAction {
+    Search {
+        query: String,
+    },
+    #[serde(other)]
+    Other,
+}
+
 #[derive(Debug, Clone, Serialize, Deserialize, PartialEq)]
 #[serde(tag = "type", rename_all = "snake_case")]
 pub enum ReasoningItemReasoningSummary {

codex-rs/protocol/src/protocol.rs

@@ -443,6 +443,8 @@ pub enum EventMsg {
 
     WebSearchBegin(WebSearchBeginEvent),
 
+    WebSearchEnd(WebSearchEndEvent),
+
     /// Notification that the server is about to execute a command.
     ExecCommandBegin(ExecCommandBeginEvent),
 
@@ -675,6 +677,11 @@ impl McpToolCallEndEvent {
 #[derive(Debug, Clone, Deserialize, Serialize)]
 pub struct WebSearchBeginEvent {
     pub call_id: String,
+}
+
+#[derive(Debug, Clone, Deserialize, Serialize)]
+pub struct WebSearchEndEvent {
+    pub call_id: String,
     pub query: String,
 }