LOL! Just today I accidentally dropped a knife directly on my finger and cut it pretty bad. I'm now unsure if Duo can tell the future, or CAUSES the future in some sort of Voodoo scenario. Either way, spooky stuff :D
When using a knife, "å skjære" would imply that the knife was either moved back and forth, like a sawing movement, or carving something. "Å kutte" refers to more of a chopping movement; cutting more or less straight down.
I guess you'd usually say "Jeg har kappet av meg fingeren" (I've chopped off my finger), as "Jeg har kuttet av meg fingeren" (I've cut my finger off) makes it sound like you've spent a few hours meticulously mutilating yourself.