ere's my taxonomy of storage:

NAS: File-based protocol access (SMB/NFS) with perhaps iSCSI block access. Largely locked to the LAN based on the nature of access protocols.

Private Cloud: Usually NAS plus some kind of HTTP/S access. May or may not include a Dropbox-type client for local synchronization. Largely limited to LAN by LAN/FW configuration itself, not the software, although I have seen some brain damaged web access components that would make use over the internet dumb or frustrating.

Private Cloud web accessible: The above, but with a hardware vendor service in the middle providing a broker service to access the device from non-LAN location.

Private Public Cloud: Yes, a contradiction in terms but a step above. Usually privately controlled hardware or VM accessible from the internet without reliance on third party broker services. Could be hosted at home, run on AWS, etc, but all software and OS is under user control.

Public Cloud: Third-party provided service, often only web accessible and with OS client for local file sync. Infrastucture is shared. Dropbox, Google Docs, etc.