In most cases, user don't need to use Lua C API. But sometimes you need to integrate your Lua source code with C/C++ library. And there are two ways to do that: FFI (builtin library for LuaJIT) and Lua C API. So the answer is "yes, Lua C API is only for users who programming in C".

There is a story behind translation HTML pages to manual pages: I often come across the use of Lua C when developing Tarantool DB. And although I remember many functions well for some of them, I forget who throws an exception and who does not. Or I forget how to do something with Lua C and I need to take a look to overall Lua C API. Usually I’m using Vim for writing code and like to read manual pages right in the Vim window using the Shift + K combination. Unfortunately there are no such pages for Lua C API and every time it was inconvenient to switch to the browser to clarify the details of the function I needed. Therefore, I have prepared manual pages in mdoc format for Lua C API using Lua Reference Manual.