DX11 not detecting GFX card

[worldofneil]

when I switch to DX11, it uses the integrated graphics from Intel

It sounds like your system is auto detecting whether or not to use the NVIDIA GPU and in this case it's only recognising the standard DX9 version (ffxiv.exe and not ffxiv_dx11.exe) as a game hence that gets to the use NVIDIA card, but the DX11 version isn't recognised as a game so it just gives it the default Intel card. First of all it's probably just worth updating your NVIDIA drivers as newer drivers might fix the autodetect/update it.

Failing that though, to fix this normally you just load up the NVIDIA Control Panel and then near the top left select "Manage 3D settings". Assuming ffxiv_dx11.exe isn't already in the list you then want to add it and look for "preferred graphics processor" or something like that so you can force the NVIDIA instead of letting it autodetect. You may also need to do the same for ffxivboot.exe although I wouldn't think so.