In other words, if you're looking to replace/supplement your C use: go Rust. If you're looking to replace/supplement your C++ use: probably go D, with some caveats towards actual use.
Note: I like both languages, but do prefer Rust for my personal use cases.
Some details here... every language other than assembly languages has some amount of runtime. This is Rust's: https://github.com/rust-lang/rust/blob/master/src/libstd/rt.... (as you allude to, many people refer to this amount of runtime as "no runtime" since it's very, very small.)
You could also consider some other things as part of a runtime; for example, by default on most platforms, Rust includes jemalloc. That can be removed, though not in stable Rust. Same with the standard library code, which can be removed for libraries, but not binaries just yet (due to a small technicality regarding the stability of some attributes.) Things that do this are likely to use some of Rust's other unstable features at the moment, so it's not generally a burden to do so, and those interfaces rarely change right now.
How are you defining "runtime" such that this statement is correct?
I see a distinction between a runtime library and a language runtime. The former is just a set of convenient routines for interacting with a specific platform. The latter is something that runs alongside your program to facilitate its execution, like a JIT or an interpreter.
The key is, you are doing the injecting in asm.
Anyway, your distinction is pretty solid. Most people just use "runtime" to mean either one, and then rely on the context to disambiguate.