[App] Improve PythonServer info message on startup (#15989)

* change msg
* update chgl
* show the user's class name

src/lightning_app/CHANGELOG.md

@@ -47,6 +47,9 @@ The format is based on [Keep a Changelog](http://keepachangelog.com/en/1.0.0/).
 
 ### Fixed
 
+- Fixed `PythonServer` messaging "Your app has started" ([#15989](https://github.com/Lightning-AI/lightning/pull/15989))
+
+
 - Fixed `AutoScaler` raising an exception when non-default cloud compute is specified ([#15991](https://github.com/Lightning-AI/lightning/pull/15991))
 
 - Fixed the debugger detection mechanism for lightning App in VSCode ([#16068](https://github.com/Lightning-AI/lightning/pull/16068))

src/lightning_app/components/serve/python_server.py

@@ -300,5 +300,7 @@ def run(self, *args: Any, **kwargs: Any) -> Any:
         fastapi_app = FastAPI()
         self._attach_predict_fn(fastapi_app)
 
-        logger.info(f"Your app has started. View it in your browser: http://{self.host}:{self.port}")
+        logger.info(
+            f"Your {self.__class__.__qualname__} has started. View it in your browser: http://{self.host}:{self.port}"
+        )
         uvicorn.run(app=fastapi_app, host=self.host, port=self.port, log_level="error")