Methinks somebody had certain things he was supposed to say. Vulkan is probably the way to go, though.
direct3d is part of directx. as is directsound (now xaudio), directinput (now xinput) and so on ... ... when people talk about dx12 they usually mean just d3d 12, but anyway as long as we understand eachother its all good isnt it vulkan will be available on nvidia and amd cards so who cares? (unless you are a developer)
But than again that would only tell half the story, it's more like Vulkan and all the OS' versus D3D 12 and win 10, since Dx12 is one of the few reason to switch to win 10 as soon as Dx12 is released. Or for Dx12 if oyure a gaming company.