Runtime describes software/instructions that are executed while your program is running, especially those instructions that you did not write explicitly, but are necessary for the proper execution of your code.
Low-level languages like C have very small (if any) runtime. More complex languages like Objective-C, which allows for dynamic message passing, have a much more extensive runtime.
You are correct that runtime code is library code, but library code is a more general term, describing the code produced by any library. Runtime code is specifically the code required to implement the features of the language itself.
Runtime is a general term that refers to any library, framework, or platform that your code runs on.
The C and C++ runtimes are collections of functions.
The .NET runtime contains an intermediate language interpreter, a garbage collector, and more.