Almost right, Think of CC as you would the CS as apps that get installed on your computer. The difference is the payment method and it now includes a cloud drive for your files. In other words CC is just a full upgrade from CS6 it would have been CS7 and now CS8 but they wanted to play on the cloud word as everyone is doing it, they have have to as well.
None of the apps run on the cloud all of your data files get saved on your own system, but you can upload them to the cloud if you choose to.