I have heard about things like \"C Runtime\", \"Visual C++ 2008 Runtime\", \".NET Common Language Runtime\", etc.
In my understanding runtime is exactly what it means - the time when the program is run. You can say something happens at runtime / run time or at compile time.
I think runtime and runtime library should be (if they aren't) two separate things. "C runtime" doesn't seem right to me. I call it "C runtime library".
Answers to your other questions: I think the term runtime can be extended to include also the environment and the context of the program when it is run, so:
This answer is to some extend just my opinion, not a fact or definition.