To answer your question, I think it's important to establish some sort of working definition of what "cloud computing" is. Here's a definition by the National Institutes of Standards and Technology, which has become more or less generally accepted: "Cloud computing is a model for enabling convenient, on-demand network access to a shared pool of configurable computing resources (e.g. networks, servers, storage, applications, and servers) that can be rapidly provisioned and released with minimal management effort or service provider interaction."

For techies, cloud computing means shared infrastructure, software or platforms delivered as a service, over an IP network (such as, but not limited to the public Internet), from a third party data center on a subscription basis. For everyone else, it means anything accessed via the Internet, which isn't exactly accurate. The lines have been blurred as companies attempt to promote some basic web services, or individually licensed software as cloud services, which isn't always the case.

With regard to public versus private cloud, both are legitimate types of cloud computing. The private cloud model is often seen as more secure, though the public model is usually accompanied with encryption and VPNs to accommodate most basic security needs. There's even a third category called "community cloud" which is more of a hybrid of the two.