I agree that it isn't that hard. But your definition doesn't quite do it. As written, your definition includes things that have died and things that aren't yet alive.

Dead plants and animals still have usable DNA for a while after death. If the tissues are preserved, sometimes it's a long while. And mostly, we don't think of spores or viruses as living things, but they certainly have DNA.

I would supplement your definition by saying that we refer to an organism as alive when its pieces are functioning in a coherent and mutually-dependent way to keep the organism as a whole alive. (Some fuzziness creeps in when you try to distinguish a composite organism from a collection of separate components...but I don't think that causes problems in practice.)