Why is "cutoff" called "idle cutoff" in some shutdown checklists?

The mixture control is attached to the throttle body and in the idle cutoff position, diverts all fuel received back to the pump inlet or aircraft fuel supply. In this manner, no fuel can flow toward the fuel injector nozzles or carb.

I think i do not have to explain what cutoff means, as others have already done that. However, it usually says that to remind you to idle the engines before cutoff. There have been a few cases were pilots did not completely idle the engine. When they come back to the plane, turn it back on, the plane can lurch forward in an instant. It's just there to remind you.

Weird phraseology.

Most engines (turbine and piston) will stop the fuel flow with the throttle/power/thrust lever in any position other than idle when the mixture/condition lever is moved to idle cutoff. It's not recommended though.

Some small business jets use the thrust levers to cutoff fuel by pulling them up through a gate and rearward (like reverse on a turboprop) so I guess the term "idle cutoff" would work in this case.

It is very simply worded different in some instances. Cutoff happens at idle in any checklist. Some call it idle cut off some call it cutoff most mechanics call it ICO. It is a different way of saying the same thing.

Cutoff during shutdown checklist is when you lean the mixture to where the fuel is stopped by the carburetor, causing to finish burning the very last bit of fuel in the cylinders (of engine), resulting the engine to stop.

Because the engine throttles are in the idle position preparatory to shutdown.

That's is because you are suppose to have the engine running at idle to stop it. Jet engines are stopped with the throttle and its just as you pass through idle going back to the stop. on the receipting engines you put the throttle in idle and then pull the mixture to idle cut off.

Because you should be at idle when you cut the fuel.

cutoff relates to fuel shutoff