I never understood the insistence in this. A transpiler is "a compiler that targets a high-level language." It's a term that adds information above and beyond compiler. All transpilers are compilers, not all compilers are transpilers.
You say it doesn't make sense, and yet you were able to figure out how to apply it correctly to the example you gave, so you have to have some understanding of the idea.
Like, in most American homes, there's a bathtub with a showerhead in it. When we're using the showerhead, we'll say, "I was in the shower..." When we're using the faucet, we'll say "I was in the bathtub." This sort of thing happens all the time. It makes plenty of sense that something is a transpiler when it's transpiling and not when it isn't.
But isn't that what this is? I was under the impression that a compiler turns source code into machine code whereas a transpiler turns source code into differnet source code?
All compilers turn one form of code into another. Whether that's C to assembly (to machine code), or JavaScript to bytecode to machine code, or Python to Python. Transpiler is just a term referring to a compiler taking one high-level language to another. It's unnecessary because a Coffeescript to JavaScript compiler is a transpiler already; saying "transpiler" there tells you nothing you don't already know.