📝 align async lifecycle examples

docs/api/async-logger-shutdown.md

@@ -47,7 +47,7 @@ Here are some specific examples provided.
 
 When a service should stop logging only after queued records are drained:
 ```moonbit
-await logger.shutdown()
+logger.shutdown()
 ```
 
 In this example, the logger waits for normal drain behavior before final closure.
@@ -56,7 +56,7 @@ In this example, the logger waits for normal drain behavior before final closure
 
 When teardown should prefer speed over preserving backlog:
 ```moonbit
-await logger.shutdown(clear=true)
+logger.shutdown(clear=true)
 ```
 
 In this example, pending work is abandoned intentionally so shutdown can complete sooner.