Wow. I am looking at the April 2018 CUDA C Programming Guide. Everything it is says “C with minimal extensions”. There is no mention of C++ and wrapping everything in ‘extern “C”’ to appease the compiler front-end. What a confusing mess!
I agree it is confusing. No idea why references to C were not summarily replaced with C++ when the compiler switch was made back in 2008, which resulted in CUDA being C++ based (it involved adopting first the Open64 then the LLVM toolchain). You may want to file RFEs against the docs where clarification is needed.
The basic libraries (CUBLAS, CUFFT, etc) that ship with CUDA use C interfaces [or at least used to, I haven’t checked recently] for easy interoperability with other languages that support such bindings, in particular Fortran, which was still fairly heavily used in science and engineering and needed to be supported right off the bat when CUDA first shipped.