Access: Many argue what makes a cloud public is who has access to the cloud. However the cloud doesn't really exist until something is provisioned. Once that thing (service, server, whatever) is provisioned you have a cloud but you also necessarily have restrictions on who can access the server. Amazon AWS, Microsoft Azure, all the vendors require logins and keys and such to protect their systems. And it's always possible to open up the gates and allow external entities to access a cloud within a company's data center. So really the differentiation cannot be access.
Security: Some argue its the security model and domain that make a public cloud. Public clouds are more accessible and therefore are more at risk. I point to all the penetrations at large companies by hackers to demonstrate at best there is no difference.
Multi-tenant: Many argue public cloud is multi-tenant. How is this different if its my competition or my sales department when the virtual machines and storage are all partitioned and sit in separate security domains? Again I see this as more similar between private and public clouds and less different.
The benefit of differentiating cloud models by who owns the assets is it plays into the focus of CFO's, covered in an earlier post, to drive IT to leverage other people's assets. IT needs move from a capital/asset intensive environment to one where costs are expensed and vary with consumption.
Therefore in my humblest of opinions using my point of differentiation, everything will move to public cloud. Why? The benefit of leveraging other people's assets!