when I worked in the violin shop many years ago, I'd occasionally get nicked by a chisel or the like when doing fine trim work. There was just no way to keep track of where i was cutting than to keep a finger from the other hand "in the way". I never get so casual with p[ower tools. I used to use a product called "New Skin" instead of bandaids, I don't know if it's still available, and have no idea if it's actually safe, but it's probably some kind of plastic dissolved in alcohol with a brush applicator in the top of a small bottle. It burns like H--- when it hits the open cut, but then you know you've applied it where it s needed. If applied over and around the cut pretty well, it forms a coating that keeps the cut closed and clean, assuming you wash first, for maybea day.
At least you don't have a messy bandaid comming off and getting caught up in things. tom Fowle