Also VKRay and DXR seem to be mentioned as competing with each other.
Yes, VKRay is Vulkan extensions for RT. So far only nvidia has had any RT to speak of, so they have defined extensions, which are likely to be brought into the Vulkan standard.
Vulkan is a competing API to DirectX, which has DXR, yes. It is cross platform and aims to be just as hardware agnostic as DirectX, but also cross-platform.
DXR is a standard defined by microsoft, there are no 'nvidia extensions' to this, the nvidia drivers provide a DXR implementation using nvidia RTX hardware at the back end.
So how do other GPUs which don't share the same design work then?
Well, they provide their own drivers which also fulfill the DirectX DXR API. It's not proprietary nvidia extensions to the API that are suddenly going to be made obsolete when it's standardised or when AMD come along. Each just provides their own drivers to fulfill the DX API requirements.
Similarly with Vulkan, there are RT extensions to the API (called VKRay), and drivers will be made to fulfill the requirements of the API.
The RT extensions are nvidia-only right now, but they aren't that way because nvidia went ahead and just made their own proprietary thing separate to what Vulkan had defined - they are made that way because Vulkan didn't have an RT API at all, so nvidia went ahead and added them. It looks like Intel and AMD will feed into standardising this, and Khronos will eventually publish that standard, but there's no reason to think it will be *very* different from what nvidia have already defined, or that somehow nvidia have been evil and written proprietary extensions to things.