I think I found a definitive solution, because I've seen so many cases ( when I grade my students' homework ).
I've never made such a case on my own, but many of my students' homework appeared to have that case of missing c/c++ category.
In those cases, build succeeded but executables not generated.
Adding new c source file to the project will display the c/c++ category, but if you remove the source file from the project, then c/c++ category will go away again.
It was not .vcxproj file problem.
I think for some reason, VS failed to recognize the source as a c source file.
So, my solution is;
Select the source file of the project in solution explorer, then right click to select the property of the c source file.
Then select 'Item Type' to 'C/C++ compiler'.
Then c/c++ category will appear.