Projects for Apple platforms usually also use .h, where it could mean anything from C/C++ to Objective-C/C++.
In practice, Clang handles mixed C/C++/Obj-C codebases pretty well and determining the language for a header never really felt like an issue since the API would usually already imply it (declaring a C++ class and/or Obj-C class would require the corresponding language to consume it).
If a C++ header is intended to be consumed from C, adding the usual should alleviate the name mangling issues.
In both cases, checkout updates your working tree (by checking out either all or just some files from a commit), just when you’re switching branches it moves your HEAD pointer too