Self Injury does not always have to be just cutting. But I wouldn't say that breaking a door would be considered self injury just because you didn't do it with the purpose of harming yourself. And if people ask about the cuts on your hand, just tell them that you were trying to fix a lawn mover (just kidding, maybe choose a piece of equipment that's actually in season right now).
Mothers can sometimes bring out the worst in us, but please know that you don't have to take it out on yourself. You're going to need those lovely hands for moving the lawn when summer comes around