File/New Project should not bring up websites at all. On mine, the
options are VB projects, including Windows Application, Console
Application, and Class Library. There are C# with the same. There's
also a Business Intelligence section, and one for Smart Devices, and
Other Project Types which are setup types.

It depends partly on what you've installed obviously. I've also
installed the Ajax templates, but they don't show up unless I specify
opening a Website, not a Project.

Hope this helps.