The English “He cut himself into his finger with the knife.” is not correct. You can say “He cut himself in the finger.” and you can say “He cut into his finger.” but it is unnatural to say “He cut himself into his finger.” Furthermore there are two additional problems here.
One, in English you typically need to say “his finger”, and it comes across as unnatural to say “the finger”. If you say “He cut the finger.” this would imply that it was not his finger, like the mental picture I get is a dismembered finger on the table, and he cuts it with a knife.
Two, if you say “He cut into his finger.” this implies intentionality, like he is TRYING to cut his finger, like perhaps he is trying to get a splinter out or something. Whereas if you say “He cut his finger.” this implies that it was an accident, like he slipped and cut his finger unintentionally.
I’m not 100% of the connotation of this German sentence, but if it’s connoting an accident, like he slipped, I think a better translation of this would be: “He cut himself in the finger with the knife.”