True enough, these terms are vague. To me a "private" cloud is one that runs on-premise behind the firewall. SO can I construct the application to run privately, then when capacity is tight, move the application onto Azure. Unless I misunderstand their technology, Eucalyptus is touting this feature for the Amazon platform.
I like the idea of developing, testing, and running locally then moving to the cloud when conditions dictate.
Actually, the way you define it, that is closer to what a lot of folks would call 'cloudbursting'. There are many ways of doing it. One example on Windows Azure I talk about often is RiskMetrics. These guys do some serious math computation on Windows Azure. They have a hybrid on-prem/cloud model and they do a great job of moving data from on-prem when they need to do a job, spinning up instances, crunching on the data and then spinning down the instances.
The tricky issues with cloudbursting are with data. If you have an app that works on-prem, it probably has close ties to data located on-prem as well. If you're moving that to the cloud, you now have some options to consider. Do you reach back in on-prem (in which case, you have latency issues especially for interactive apps) or do you do some form of data replication?
As far as a private cloud with Windows Azure is concerned, check out the Windows Azure appliance which we just announced. That gives you an idea of the direction we're headed in