Can't this also mean that he cut off his thumb?
Yes, in this context where the "c'est horrible" exclamation seems to be a bit excessive for a simple cut.
SItesurf, is there a way in French to distinguish between a simple cut and cutting it off?
I'm wondering the same thing.
Apparently the only way to do it would be using the verb amputer, because both cut (EN)/schneiden (DE)/ciąć (PL) and cut off (EN)/abschneiden (DE)/odciąć (PL) translate to couper (or actually se couper in this instance).