This may be surprising to the untrained mind, but you don't need a definition of x to prove that x doesn't exist. All you need is to know one or more properties of x, and then you show that these properties contradict one another. Here the properties associated to God are omnipotence and goodness. If you don't believe that these are properties of God, I kindly invite you to refer to a dictionary. It's also true that the problem of evil argument assumes that good agents want to prevent bad things, which in turn implies that good things want to affect the universe. But again, any conception of morality in which good agents don't want to prevent bad things is deliberately using the words "good" and "bad" in a perverse manner. Finally, let me point out that the law of unintended consequences holds no terror for an omnipotent being: if bad things happen latter on, It can just fix them, too.