What is the difference between an engine "shut down", "cut-off" and "cut down"?
To be specific:

Shutdown: To take a step or series a steps to stop the operation of a machine and put it into its normal inoperative state. A shut-down procedure may include cutting off fuel, closing valves, and turning off pumps, or may just be flipping a switch to OFF.

Cut-off: To interrupt (or a noun describing something that interrupts) the supply of something to a machine; fuel, water, electricity.  Generally refers to closing of a valve or opening an electrical contactor or switch.  On a piston engine, you have a mixture setting called "idle cut-off" that blocks flow of fuel at the carburetor. You might see a switch that opens a contactor to an electrical bus, for use in emergencies, labelled "Bus XXXX Cutoff".

Cut-down: I've been working in aviation in N America since 1975 and have never heard of this term as relating to operation of an engine (maybe in UK?).  Normally if I see the word cut-down it's an adjective that describes something that has been physically altered to make it shorter or smaller.  Like a cut-down vertical fin, or a cut-down antenna, or cut-down propeller blades.  I'd be surprised if anybody on here has ever heard of the phrase "cut down power" as an instruction to reduce power, or thrust.  You just say "reduce".

You DO sometimes hear the word "Cut" by itself as an instruction to go to idle or to remove power completely on an engine.  A skydiving pilot may get an instruction to "Cut" when it's time for jumpers to exit.  Some day when you are on a jet type course, one of the simulator sessions will include "V1 Cuts", where one engine is failed just as you are passing V1 (whereupon hilarity ensues the first time you do it).  So in those cases, power is cut, but not "cut-down".