Windows only allows you connect to a NAS with a single set of credentials. You cannot ‘mix and match’ credentials to access different user shares; the one user you pick to use must have access rights to any Share you wish to map.
If you connect to Public first, and don’t enforce the ‘connect as other user’, it will connect and associate your Windows user credentials with the NAS. Thus, you will not be able to connect to a Private MyCloud share that does require credentials.
That’s why you need to disconnect all mapped drives, reboot the PC, and then re-map shares, forcing it to use your MyCloud user credentials.