Docker on Windows is a way to refer to just the Docker Engine running on Windows. I find it helpful to think of this as a Windows Container Host, so yes Windows containers only. This would be what you would run on a Windows Server 2016 machine. So maybe a better name is Docker for Windows Server which I believe people have used as well. I still prefer a Windows Container Host. Which means it only has the Docker Engine at the end of the day, doesn't even need to have any of the Docker clients.
Docker for Windows is a product meant for running both Linux and Windows containers on Windows. It's not meant for a production environment, and instead is meant for a desktop/client SKU of Windows, hence the Windows 10 requirement. So, you could think of this as Docker for Windows 10. Because Docker for Windows can run both container types.