There was a huge change in the architecture of DirectX when Vista came out. All versions of Direct3D up to 9 were pretty much all-in APIs. With Direct3D 10, part of the API -mostly all the low level stuff like talking to the hardware- was sent to a different API called DXGI. Direct3D10, 10.1 and 11 are different services that interact with DXGI.
Even though they share the same name, Direct3D 9 and below are completely different APIs to DXGI, and that's why you can have both working side to side on Vista and above (you couldn't have Direct3D8 and 9 installed on the same computer. In fact, Direct3D9 is AFAIK backwards compatible a long way back)
Regarding the caps viewer, "DXGI" makes reference to what your card is available to do when running a DXGI-based program (Direct3D 10 and above), while "Direct3D9" makes reference to what your card is available to do when running a program based on Direct3D 9. That's all there is to it.
You see that
PixelShaderVersion is 3 in the Direct3D9 device, because that's the last version supported by Direct3D 9. You cannot use Shader model 4.0 from a Direct3D 9 program, even if the card supports it.
Regarding DXGI support, if your card supports Direct3D9, then it supports DXGI, and therefore Direct3D11. Direct3D11 is not based on "caps" like 9 and before, but on feature levels. The most basic feature level (9_1) is (mostly) equivalent to Direct3D9 support, so any program written with Direct3D11 that targets feature level 9_1 will most likely work on a Direct3D9 capable card, even if it came out before Direct3D11 came out.
Remember that "feature level" is different from the Direct3D version: if the caps viewer says that your card supports up to shader model 4.0, then it probably means that it is unlikely to support feature levels above 10_0, but you can certainly create a Direct3D11 device based on feature level 9_1 if you want.
Supporting feature level 11_0 and above is what manufacturers usually mean with "Direct3D11 support".
Regarding DirectDraw, that's a completely different, and even older (DirectX 7 era) API.
You can read more about DXGI in here, and feature levels in here. The docs are your friends.