fix: propagate recording session to screencast stream

After `record start`, the active page switches to the new recording
context but `update_stream_client()` is never called. This causes the
live preview (screencast) to stay on the old pre-recording context
while CLI commands execute on the new recording context.

The same issue affects `record stop` — the screencast doesn't sync
back to the current active page.

Fix: call `state.update_stream_client().await` in both
`handle_recording_start` and `handle_recording_stop`, matching the
pattern used by all other commands that switch contexts (open, close,
connect, tab switch, etc.).

This is the same class of bug as #1019 (downloads not working in
recording context) — state that needs to be propagated to the new
recording context.

Author: shtefcs

cli/src/native/actions.rs

@@ -3711,6 +3711,11 @@ async fn handle_recording_start(cmd: &Value, state: &mut DaemonState) -> Result<
         server.set_recording(true, &state.engine).await;
     }
 
+    // Propagate the new recording session to the screencast stream.
+    // Without this, the live preview stays on the old (pre-recording) context
+    // while CLI commands execute on the new recording context.
+    state.update_stream_client().await;
+
     Ok(result)
 }
 
@@ -3722,6 +3727,10 @@ async fn handle_recording_stop(state: &mut DaemonState) -> Result<Value, String>
         server.set_recording(false, &state.engine).await;
     }
 
+    // Sync the screencast stream back to the current active page
+    // (which may have changed during recording).
+    state.update_stream_client().await;
+
     result
 }