You write “the 2nd law applies to any system, whether open or closed.”.

Yes, the 2nd law does apply to refrigerators. Specifically, the 2nd law is what guarantees that refrigerators produce a net positive amount of entropy while they are running. This is why a refrigerator heats up the room it operates in. It cools the temperature inside of the fridge while releasing the excess heat out the back as exhaust. You are correct that the entropy inside decreases. But this is only because it is releasing entropy out into the room as heat — and because of the 2nd law it always has to give off more entropy as exhaust than the amount of entropy reduction that happens inside.