(I'm not sure if you'll see this or not, but it's worth a shot)
Like I said, before a compiler can be certified ANSI C, it must provide the Standard C Library (for whatever platform the compiler is for). As a programmer, you don't have to use the supplied functions, but they are there, and they do comprise a part of the C Standard. And because the functions are defined, the compiler writer can do some pretty neat things.
For instance, include string.h, and that informs the compiler you want to use the string and memory functions defined by the C Standard. The compiler can then generate code directly for, say, memcpy() instead of generating a call to said function (it can inline it, even though C89 made no standard method of inlining a function). Or the function sin() if you include math.h (some CPUs support a single instruction for the sin function). Don't include string.h, and well ... what happens is up to the compiler. Most just give a warning about an undefined function, assume it's defined as "int unknownfunction()" and leave it up to the link phase to either find it or not.
Yes, there is a distinction between the C language, and the C library, but both must be provided if a C compiler wants to conform to the C Standard.