Yes, you should be able to tell your direct manager basically what you told us. Copy and paste into an email if that’s easier. (Your tone here is fine.)
It does sound like your manager’s manager has poor people skills / favors a more confrontational management style. If you have a good direct manager that can be tolerable, up to you to decide how much it bothers you.
It does sound like your manager’s manager has poor people skills / favors a more confrontational management style. If you have a good direct manager that can be tolerable, up to you to decide how much it bothers you.