So do I, or actually, it is named "Public"
A user share named "internal" is at /mnt/user/internal. Any top level folder named "internal" on cache, or on any array disk not excluded in Global Share Settings, is part of the "internal" user share. So, /mnt/cache/internal, /mnt/disk1/internal, /mnt/disk2/internal ... are all part of the user share named "internal".
A user share named "public" is at /mnt/user/public. Any top level folder named "public" on cache, or on any array disk not excluded in Global Share Settings, is part of the "public" user share. So, /mnt/cache/public, /mnt/disk1/public, /mnt/disk2/public, ... are all part of the user share named "public".
So, as you can see, there is no way to have a user share named "internal" that refers to /mnt/cache/public, /mnt/disk1/public, (skip /mnt/disk2/public), ...
And anyway, I suspect that disk that you want to avoid accessing already contains several top level folders, and so already contains parts of several user shares. Each of those top level folders on that disk is part of the user share with the same name as that top level folder.