I put "the cat is not running fast" which is essentially the same core idea, but how far off am I with that answer, really? I get how it is "the running cat is not fast" but I'm curious.
It wants you to translate the verb describing the cat. Sure the ideas are similar enough but this grammar point is the focus.
In the original sentence, running is an adjective describing the cat, whereas in your sentence, running is a verb. In terms of meaning, the original sentence is identifying a cat (presumably among a group of cats) with the identifying feature that it is currently running, and then making the remark that it's not fast (potentially not fast in general, not just at this moment), whereas your sentence is just saying that the cat (which should be obvious to the reader given context, e.g. there's only one cat) is currently not running fast.
It's a fairly subtle distinction and in many cases they're equivalent, but not always. The core of the difference is the function of running as an adjective vs a verb and the function of fast as an adjective vs an adverb.
So again in the original, running and fast are adjectives describing the cat, whereas in yours running is a verb and fast is an adverb describing running.
It means the same thing but it's a different sentence structure. Your sentence would look something like 고양이가 빠르게 달리고 있지 않아요.