docs: add note that startup banner displays before async tasks begin

Author: Darshan808

docs/source/developers/extensions.rst

@@ -87,6 +87,10 @@ Here is basic example:
         """
         await asyncio.sleep(.1)
 
+.. note:: The server startup banner (displaying server info and access URLs) is printed before starting asynchronous tasks, so those tasks might still be running even after the banner appears.
+
+..  WARNING: This note is also present in the "Starting asynchronous tasks from an ExtensionApp" section.
+   If you update it here, please update it there as well.
 
 Making an extension discoverable
 --------------------------------
@@ -386,6 +390,11 @@ Here is a basic (pseudo) code example:
         async def stop_extension(self):
             self.my_background_task.cancel()
 
+.. note:: The server startup banner (displaying server info and access URLs) is printed before starting asynchronous tasks, so those tasks might still be running even after the banner appears.
+
+..  WARNING: This note is also present in the "Starting asynchronous tasks from an extension" section.
+   If you update it here, please update it there as well.
+
 
 Distributing a server extension
 ===============================

jupyter_server/serverapp.py

@@ -3125,7 +3125,7 @@ def start_app(self) -> None:
                         "    %s" % self.display_url,
                     ]
 
-                self.log.critical("\nDP_DAI2\n" + "\n".join(message))
+                self.log.critical("\n".join(message))
 
     async def _cleanup(self) -> None:
         """General cleanup of files, extensions and kernels created