In my library I have two entries for Shadow Warrior 2013, and you can guess what they are from reading the title of this thread. So, as far as I've been able to determine from reading around, the DirectX 11 version is the up to date version of the game, and the other is the original that stopped getting updates when they released the DirectX11 version here.

Is that right? Because if that is accurate, I can't see any reason why the larger out of date version even exists...........

The DX11 version has some changes to the game, not only visually but it adds some areas and a boss fight or two, so there's the older version not only if you have an older computer that doesn't use DX11, but want the version without those changes. Personally I think the new stuff is good, so in my mind it's really for the people with older computers.

Why would Shadow Warrior 2 not start with error that says I need directx 11 when I am running windows 10?
This is the second GOG game I have not been able to play due to this issue.

Because either your graphics card doesnt support the required DX11 level. Or your graphics card drivers are not installed properly. Or your DirectX installation is flawed.

Has nothing to do with GOG.