The principle ties liability to the course of employment, so the real question is not who Y normally works for, it is who Y was working for, and under whose direction, at the moment of the accident.
Ordinarily Y works for the caterer, but the facts show X stepping outside that arrangement, X personally asked Y to go fetch a mechanic and personally promised to pay Y 1000 for doing it. That direct instruction paired with a direct promise of payment is what creates an employment relationship for that specific task, separate from Y's regular job with the caterer, and it is X's errand, not a catering duty, that Y was carrying out when he got hurt.
Once the task and the relevant employer for that task are identified as X, the general facts that X organized the party or that Y is normally the caterer's employee stop being decisive, since neither one addresses who was actually directing Y and paying him at the moment the injury happened. That points liability at X specifically for this errand.
So the correct answer is X is liable as Y was working in the course of employment offered by X.
Asking what fact would need to be different for each option to hold makes clear why only one survives.
Since X's direct instruction and promised payment already establish the relevant employment relationship for this errand, and the other three options rest on facts that are not actually present, X remains the party liable.
Therefore, the correct answer is X is liable as Y was working in the course of employment offered by X.