refactor: improve notification handling

- Always validate all notifications to ensure proper model structure
- Handle cancellation notifications separately (no forwarding to stream)
- Simplify notification type checking and error handling
- Improve code structure and documentation

Author: donghao1393

src/mcp/shared/session.py

@@ -289,15 +289,11 @@ async def _send_response(
             )
             await self._write_stream.send(JSONRPCMessage(jsonrpc_response))
 
-    def _should_validate_notification(self, message_root: JSONRPCNotification) -> bool:
+    def _is_cancellation_notification(self, message_root: JSONRPCNotification) -> bool:
         """
-        Determines if a notification should be validated.
-        Internal notifications (like notifications/cancelled) should be ignored.
+        Determines if a notification is a cancellation notification.
         """
-        return (
-            getattr(message_root, "method", None) != "notifications/cancelled" and
-            not self._closed
-        )
+        return getattr(message_root, "method", None) == "notifications/cancelled"
 
     async def _receive_loop(self) -> None:
         async with (