Below is my take of the three phrasal verbs of cut:
- cut off a finger means your finger is separated from your hand as if by a kitchen knife while you prepare for dinner.
- cut up a finger means you’ve already got a human finger, for example, in a lab, and you chop it up into tiny pieces to do experiments on(like what you’d do with onions?)
- for cut down.. well I feel cut down a finger = cut it off but it doesn’t seem to be right to me? Maybe cut down a finger just doesn’t make any sense and cut down is more used for quantities like cut down in calories?
Btw, if you know of any other phrasal verbs of cut that can be used for fingers don’t hesitate to bring them up!