cut back on something means consume less (e.g. My doctor wants me to cut back on sweets and fatty foods.)
cut down (on something) means eat, drink or do something less often (e.g. I’m trying to cut down on coffee. I drink far too much of it)
cut in means interrupt (e.g. Your father cut in while I was dancing with your uncle.)
cut in means pull in too closely in front of another vehicle (e.g. The bus driver got angry when that car cut in.)
cut in means start operating (of an engine or electrical device (e.g. The air conditioner cuts in when the temperature gets to 22°C.)
cut somebody off means take out of a will (e.g. My grandparents cut my father off when he remarried.)
cut something down means make something fall to the ground (e.g. We had to cut the old tree in our yard down after the storm.)
cut something off means remove with something sharp (e.g. The doctors cut off his leg because it was severely injured.)
cut something off means stop providing (e.g. The phone company cut off our phone because we didn’t pay the bill.)
cut something out (of a newspaper etc.) (e.g. There was a beautiful picture in the magazine, so I cut it out and keep it)
cut something out means remove part of something (e.g. I cut this ad out of the newspaper.)