Visual Studio 2012 dropped the "Test View" window that we used to use to run tests in VS 2010. The new "Test Explorer" window only recognizes Unit Tests and Coded UI Tests, as you've discovered. This totally threw me off too.
To run web and load tests (besides the obvious but tedious method of actually opening them and clicking "Run" in the editor window) you have to select the tests you want to run in the Solution Explorer, then Load Test (menu) > Run > Selected Test. Or alternatively "All Tests in Solution".
You also have the option of running the individual Web Tests that are contained in any open Load Test. Right-click on the Test Mix node (in Scenario) or any of the tests in the Test Mix.
For completeness here is the authoritative source: http://msdn.microsoft.com/en-us/library/ms182541.aspx
By the way, here is another gotcha: although you still choose testsettings for a Load Test from the Load Test menu, you now set the testsettings for Web Test execution separately, under the Test menu: