Show activity for standalone web search calls (#24693)

## Why

Standalone `web.run` calls run in the extension, so they need normal
web-search progress activity while a request is in flight and durable
completed activity after a thread is reloaded.

Follow-up to #23823; uses the extension turn-item emission path added in
#24813.

## What changed

- Emit standalone `web.run` start/completion items through the host
turn-item emitter, preserving standard client delivery and rollout
persistence.
- Include useful completion detail for queries, image queries, and
literal-URL `open`/`find` commands.
- Render completed searches as `Searched the web` or `Searched the web
for <detail>`, with snapshot coverage for the detail-free case.
- Extend the app-server round-trip test to verify completed search
activity is reconstructed by `thread/read` after a fresh-process reload.

## Testing

- `just test -p codex-web-search-extension`
- `just test -p codex-app-server -E
"test(standalone_web_search_round_trips_encrypted_output)"`

codex-rs/tui/src/history_cell/search.rs

@@ -4,7 +4,7 @@ use super::*;
 
 fn web_search_header(completed: bool) -> &'static str {
     if completed {
-        "Searched"
+        "Searched the web"
     } else {
         "Searching the web"
     }
@@ -104,7 +104,8 @@ impl HistoryCell for WebSearchCell {
         let text: Text<'static> = if detail.is_empty() {
             Line::from(vec![header.bold()]).into()
         } else {
-            Line::from(vec![header.bold(), " ".into(), detail.into()]).into()
+            let separator = if self.completed { " for " } else { " " };
+            Line::from(vec![header.bold(), separator.into(), detail.into()]).into()
         };
         PrefixedWrappedHistoryCell::new(text, vec![bullet, " ".into()], "  ").display_lines(width)
     }
@@ -115,7 +116,8 @@ impl HistoryCell for WebSearchCell {
         if detail.is_empty() {
             vec![Line::from(header)]
         } else {
-            vec![Line::from(format!("{header} {detail}"))]
+            let separator = if self.completed { " for " } else { " " };
+            vec![Line::from(format!("{header}{separator}{detail}"))]
         }
     }
 }

codex-rs/tui/src/history_cell/snapshots/codex_tui__history_cell__tests__web_search_history_cell_snapshot.snap

@@ -2,5 +2,5 @@
 source: tui/src/history_cell.rs
 expression: rendered
 ---
-• Searched example search query with several generic words to
-  exercise wrapping
+• Searched the web for example search query with several generic
+  words to exercise wrapping

codex-rs/tui/src/history_cell/snapshots/codex_tui__history_cell__tests__web_search_history_cell_transcript_snapshot.snap

@@ -2,5 +2,5 @@
 source: tui/src/history_cell.rs
 expression: rendered
 ---
-• Searched example search query with several generic words to
-  exercise wrapping
+• Searched the web for example search query with several generic
+  words to exercise wrapping

codex-rs/tui/src/history_cell/snapshots/codex_tui__history_cell__tests__web_search_history_cell_without_detail_snapshot.snap

@@ -0,0 +1,5 @@
+---
+source: tui/src/history_cell.rs
+expression: rendered
+---
+• Searched the web

codex-rs/tui/src/history_cell/tests.rs

@@ -1084,6 +1084,14 @@ fn standalone_windows_update_available_history_cell_snapshot() {
     insta::assert_snapshot!(rendered);
 }
 
+#[test]
+fn web_search_history_cell_without_detail_snapshot() {
+    let cell = new_web_search_call("call-1".to_string(), String::new(), WebSearchAction::Other);
+    let rendered = render_lines(&cell.display_lines(/*width*/ 64)).join("\n");
+
+    insta::assert_snapshot!(rendered);
+}
+
 #[test]
 fn web_search_history_cell_wraps_with_indented_continuation() {
     let query = "example search query with several generic words to exercise wrapping".to_string();
@@ -1100,8 +1108,8 @@ fn web_search_history_cell_wraps_with_indented_continuation() {
     assert_eq!(
         rendered,
         vec![
-            "• Searched example search query with several generic words to".to_string(),
-            "  exercise wrapping".to_string(),
+            "• Searched the web for example search query with several generic".to_string(),
+            "  words to exercise wrapping".to_string(),
         ]
     );
 }
@@ -1119,7 +1127,10 @@ fn web_search_history_cell_short_query_does_not_wrap() {
     );
     let rendered = render_lines(&cell.display_lines(/*width*/ 64));
 
-    assert_eq!(rendered, vec!["• Searched short query".to_string()]);
+    assert_eq!(
+        rendered,
+        vec!["• Searched the web for short query".to_string()]
+    );
 }
 
 #[test]