In my life is a private consultant to large clients and later also as an enterprise architect at the world's best software company in my day job, I have been trying to explain to people why Private/Public/Hybrid is not fully accurate in terms of Cloud classifications. (I am not even talking about Community Cloud in this post).
My position on Cloud is that we have Public, Private (including Hosted Private). and two more... What everyone calls Hybrid is what I say really should be called Parallel, and Hybrid should be the name given to a separate type.
By Parallel I am referring to companies using Private cloud for some functions, and Public cloud for other applications. There is likely no interaction between the two. The overall benefit is a compromise between the best of both Public and Private, in a sort of "weighted average" way.
What we SHOULD be calling Hybrid would be the case of enterprise architectures that span public and private clouds for one or a set of interoperating or integrated business applications. For example, if a firm uses private cloud for running its mission critical business application, but that application seamlessly also moves specific parts of its workload to and from a public cloud. Here is a rudimentary example. A stock trading firm uses its own high performance trading applications on its own private cloud platform. But the private cloud hosted apps ship some parts of their work, e.g. less time critical analysis of large data sets, to a public cloud big data crunching services provider.
You can see that this has far deeper integration between private and public cloud uses and is clearly different from a company running all mission critical applications on a private cloud but, say, simply using outlook.com or gmail for business email. The latter is a good use of Parallel cloud computing.