Restore attachedViaVT property to distinguish VibeTunnel sessions

- Add attachedViaVT to SessionInfo interface to identify nested sessions
- Set attachedViaVT=true when VIBETUNNEL_SESSION_ID env var is present
- Add property to Swift ServerSessionInfo for API compatibility
- Allows distinguishing between direct terminal sessions and those spawned from within VibeTunnel

mac/VibeTunnel/Core/Services/SessionMonitor.swift

@@ -36,6 +36,7 @@ struct ServerSessionInfo: Codable {
     let remoteId: String?
     let remoteName: String?
     let remoteUrl: String?
+    let attachedViaVT: Bool?
 
     var isRunning: Bool {
         status == "running"

web/src/server/pty/pty-manager.ts

@@ -335,6 +335,9 @@ export class PtyManager extends EventEmitter {
       logger.debug(chalk.blue(`Creating PTY session with command: ${resolvedCommand.join(' ')}`));
       logger.debug(`Working directory: ${workingDir}`);
 
+      // Check if this session is being spawned from within VibeTunnel
+      const attachedViaVT = !!process.env.VIBETUNNEL_SESSION_ID;
+
       // Create initial session info with resolved command
       const sessionInfo: SessionInfo = {
         id: sessionId,
@@ -354,6 +357,7 @@ export class PtyManager extends EventEmitter {
         gitHasChanges: options.gitHasChanges,
         gitIsWorktree: options.gitIsWorktree,
         gitMainRepoPath: options.gitMainRepoPath,
+        attachedViaVT,
       };
 
       // Save initial session info

web/src/shared/types.ts

@@ -84,6 +84,12 @@ export interface SessionInfo {
   gitStagedCount?: number; // Number of staged files
   gitAddedCount?: number; // Number of added files
   gitDeletedCount?: number; // Number of deleted files
+  /**
+   * Whether this session was spawned from within VibeTunnel itself.
+   * Used to distinguish between direct terminal sessions and nested VibeTunnel sessions.
+   * Sessions with attachedViaVT=true are spawned from within an existing VibeTunnel session.
+   */
+  attachedViaVT?: boolean;
 }
 
 /**