It depends. Closed and private usually mean the same--hams who doesn't have permission are not permitted to use the machine. If it's listed as closed, you have to get permission to use it. If it's listed as private, don't even try to get permission--you won't.
Clubs usually have repeaters where the machines can be used, but the additional features on the machine are reserved for members only. They're not considered closed unless the club states so, and they usually don't, since that's where they get their new members.