There is still one tab on the Settings page of the Administration menu that we have not discussed yet. It's the Projects tab. So let's check it out now:
A project in Redmine can be either public or private. A public project is visible to everyone, even to unregistered users (unless you have restricted access to the whole of Redmine by enabling the Authorization required option under the Authentication tab). Of course, access to some pages of the project can still be restricted, but even if every page is restricted, the public project will still be visible! It will just appear to be empty. A private project is the opposite—it cannot be seen by non-members, whatever you do! Thus, you can let unregistered users and non-members see everything, but still such projects won't be visible to them.
So, the New projects are public by default setting should be enabled only if all of your projects are to be public. Otherwise, it is possible that a project that was meant to be private...