"It is not the waiter hurrying out, but the cook."
Translation:Nem a pincér siet ki, hanem a szakács.
There's a similar question on the facebook group: https://www.facebook.com/groups/943246729103874/permalink/1048502398578306/
So here's what's happening here: the verb here is "siet" and the grammatical place before the verb is an important place. Technically, with "kisiet" it is the "ki" that is before the verb, so if something else needs to take the place before the verb, then "ki" must go after. Now there are a few things that can do this, one is eg. negation, and another one is when there is a focus (I believe in this case the focus is a very technical expression and not just meaning the important part) and a very typical thing when you get a focus is when you have things in opposition: not the waiter but the cook. I got all this from this article: https://qubit.hu/2018/12/09/lassan-megertjuk-a-magyar-mondattant-vagy-lassan-ertjuk-meg which came out just today, unfortunately, it's all in Hungarian ...
I guessed I would be wrong with kisiet and so I was but I wanted to check. I still don't understand why the emphasis is with ki. If I was saying this in English, the emphasis would be on the identity of the person hurrying out with the fact of hurrying out being a constant. Why emphasize ki in this instance?
I'll try to find out why this is the case (typical problem: I'm a native speaker and thus right now I have no idea what the rule actually is), but: "Nem a pincér kisiet, hanem a szakács." is just something you don't say. So in this case the negation of "pincér" is the central point of the senctence and the separation of siet and ki is for some other reason ...