You’re quite right, I’m sorry it was such a hassle to track down. The two you need are the Visual C++ 2013 runtime and the DirectX runtime.
We do have an MSI package which installs these prerequisites as Dan mentions, though it’s not part of the current distribution. I’ll try to clear that to be added in to the next release, and ask the documentation team to call it out.
I believe we’re using an older version of XInput for compatibility with older versions of Windows (like XP), where it should fall back to using the OpenGL renderer if DX11 isn’t available. Microsoft don’t include the older XInput DLLs with Windows 8 by default, which is why it needs to be installed manually. You can force the game to run in OpenGL mode with the -OpenGL commandline option if you want to try it.