fix: handle all web_search actions and in progress invocations (#9960)

### Summary
- Parse all `web_search` tool actions (`search`, `find_in_page`,
`open_page`).
- Previously we only parsed + displayed `search`, which made the TUI
appear to pause when the other actions were being used.
- Show in progress `web_search` calls as `Searching the web`
  - Previously we only showed completed tool calls

<img width="308" height="149" alt="image"
src="https://github.com/user-attachments/assets/90a4e8ff-b06a-48ff-a282-b57b31121845"
/>

### Tests
Added + updated tests, tested locally

### Follow ups
Update VSCode extension to display these as well

codex-rs/protocol/src/models.rs

@@ -157,7 +157,9 @@ pub enum ResponseItem {
         #[serde(default, skip_serializing_if = "Option::is_none")]
         #[ts(optional)]
         status: Option<String>,
-        action: WebSearchAction,
+        #[serde(default, skip_serializing_if = "Option::is_none")]
+        #[ts(optional)]
+        action: Option<WebSearchAction>,
     },
     // Generated by the harness but considered exactly as a model response.
     GhostSnapshot {
@@ -1034,10 +1036,12 @@ mod tests {
                         "query": "weather seattle"
                     }
                 }"#,
-                WebSearchAction::Search {
+                None,
+                Some(WebSearchAction::Search {
                     query: Some("weather seattle".into()),
-                },
+                }),
                 Some("completed".into()),
+                true,
             ),
             (
                 r#"{
@@ -1048,10 +1052,12 @@ mod tests {
                         "url": "https://example.com"
                     }
                 }"#,
-                WebSearchAction::OpenPage {
+                None,
+                Some(WebSearchAction::OpenPage {
                     url: Some("https://example.com".into()),
-                },
+                }),
                 Some("open".into()),
+                true,
             ),
             (
                 r#"{
@@ -1063,26 +1069,43 @@ mod tests {
                         "pattern": "installation"
                     }
                 }"#,
-                WebSearchAction::FindInPage {
+                None,
+                Some(WebSearchAction::FindInPage {
                     url: Some("https://example.com/docs".into()),
                     pattern: Some("installation".into()),
-                },
+                }),
                 Some("in_progress".into()),
+                true,
+            ),
+            (
+                r#"{
+                    "type": "web_search_call",
+                    "status": "in_progress",
+                    "id": "ws_partial"
+                }"#,
+                Some("ws_partial".into()),
+                None,
+                Some("in_progress".into()),
+                false,
             ),
         ];
 
-        for (json_literal, expected_action, expected_status) in cases {
+        for (json_literal, expected_id, expected_action, expected_status, expect_roundtrip) in cases
+        {
             let parsed: ResponseItem = serde_json::from_str(json_literal)?;
             let expected = ResponseItem::WebSearchCall {
-                id: None,
+                id: expected_id.clone(),
                 status: expected_status.clone(),
                 action: expected_action.clone(),
             };
             assert_eq!(parsed, expected);
 
             let serialized = serde_json::to_value(&parsed)?;
-            let original_value: serde_json::Value = serde_json::from_str(json_literal)?;
-            assert_eq!(serialized, original_value);
+            let mut expected_serialized: serde_json::Value = serde_json::from_str(json_literal)?;
+            if !expect_roundtrip && let Some(obj) = expected_serialized.as_object_mut() {
+                obj.remove("id");
+            }
+            assert_eq!(serialized, expected_serialized);
         }
 
         Ok(())