feat(app-server): update remote control APIs for better UX (#22877)

## Why
To help improve `codex remote-control` CLI UX which I plan to do in a
followup, this PR adds `server-name` to the various remote control APIs:
- `remoteControl/enable`
- `remoteControl/disable`
- `remoteControl/status/changed`

Also, add a `remoteControl/status/read` API. This will be helpful in the
Codex App.

codex-rs/app-server/src/lib.rs

@@ -41,7 +41,6 @@ use codex_analytics::AppServerRpcTransport;
 use codex_app_server_protocol::ConfigLayerSource;
 use codex_app_server_protocol::ConfigWarningNotification;
 use codex_app_server_protocol::JSONRPCMessage;
-use codex_app_server_protocol::RemoteControlStatusChangedNotification;
 use codex_app_server_protocol::ServerNotification;
 use codex_app_server_protocol::TextPosition as AppTextPosition;
 use codex_app_server_protocol::TextRange as AppTextRange;
@@ -1003,14 +1002,9 @@ pub async fn run_main_with_transport_options(
                             continue;
                         }
                         remote_control_status = status.clone();
+                        let notification = ServerNotification::RemoteControlStatusChanged(status);
                         initialize_notification_sender
-                            .send_server_notification(ServerNotification::RemoteControlStatusChanged(
-                                RemoteControlStatusChangedNotification {
-                                    status: status.status,
-                                    installation_id: status.installation_id,
-                                    environment_id: status.environment_id,
-                                },
-                            ))
+                            .send_server_notification(notification)
                             .await;
                     }
                     created = thread_created_rx.recv(), if listen_for_threads => {

codex-rs/app-server/src/message_processor.rs

@@ -897,6 +897,10 @@ impl MessageProcessor {
                 .remote_control_processor
                 .disable()
                 .map(|response| Some(response.into())),
+            ClientRequest::RemoteControlStatusRead { .. } => self
+                .remote_control_processor
+                .status_read()
+                .map(|response| Some(response.into())),
             ClientRequest::ConfigRequirementsRead { params: _, .. } => self
                 .config_processor
                 .config_requirements_read()

codex-rs/app-server/src/request_processors/remote_control_processor.rs

@@ -5,6 +5,7 @@ use crate::transport::RemoteControlUnavailable;
 use codex_app_server_protocol::JSONRPCErrorError;
 use codex_app_server_protocol::RemoteControlDisableResponse;
 use codex_app_server_protocol::RemoteControlEnableResponse;
+use codex_app_server_protocol::RemoteControlStatusReadResponse;
 
 #[derive(Clone)]
 pub(crate) struct RemoteControlRequestProcessor {
@@ -31,6 +32,16 @@ impl RemoteControlRequestProcessor {
         Ok(RemoteControlDisableResponse::from(handle.disable()))
     }
 
+    pub(crate) fn status_read(&self) -> Result<RemoteControlStatusReadResponse, JSONRPCErrorError> {
+        let status = self.handle()?.status();
+        Ok(RemoteControlStatusReadResponse {
+            status: status.status,
+            server_name: status.server_name,
+            installation_id: status.installation_id,
+            environment_id: status.environment_id,
+        })
+    }
+
     fn handle(&self) -> Result<&RemoteControlHandle, JSONRPCErrorError> {
         self.remote_control_handle
             .as_ref()