If we look at the definition for cloud computing as formulated by NIST, then part of the essential characteristics of a cloud is location independence. This also holds true for other questions and replies on this website, it's always cloud vs. On-premises. Example However in that very same paper the description of the private cloud says:
.. it may exist on or off premises ...
But how can a private cloud be both On-premises and a cloud? This NIST definition is the basis of definitions regarding cloud computing that I see in other research papers and also used to answer other questions about cloud computing on here. So it seems to be pretty relevant and accepted to this day.
So yeah the question is in the title, how can this definition work? How can the private "cloud" be On-premises?
Yes, this is actually a popular architecture where some of the system's machines run on your premises, while the rest runs on the the cloud provider's datacenters.
It usually happens when some of the system's data is so sensitive that you cannot risk to keep it on a remote datacenter, or when some specific (often internal) applications don't afford to have a network latency, so it's kept near the clients.
Network connections between these two types of machines becomes more complex that way, but some businesses are okay with it for the benefits they get from the on-premises cloud.