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.)
Oh yeah, good call, dead things. You know, the more I think about it, the more I think the concept of "dead" might be even more fraught with vitalism than "alive" is.
Recently, Hank Green posted a video discussing the definition of life. He offered two definitions; that life acts in a manner to achieve a goal, and that life continuously decreases internal entropy.
There are problems with these definitions. The first definition includes every machine that has a function. The second one includes, for example, a machine that constantly reshapes parts of its body into a paperclip.
Other definitions of life are equally confusing, doing things like excluding viruses because they use other cells to reproduce, despite meeting the intuitive meaning we have for life.
So, dissolve "life." Why do we care if something is alive? To decide if its life has value. Hank dances around the issue, showing that life has no inherent value by using mouthwash to kill billions of bacteria in his mouth. But, he doesn't take this to its conclusion. It doesn't matter if something is alive or not. We won't suddenly care about the well being of viruses if a new definition of life comes along tomorrow pronouncing viruses to be living. What we value is sentience.
Bugs are extremely low on the sentience scale, so we feel free to kill them. Animals that are higher on the scale, such as cats, have laws preventing any sort of mistreatment.
tl;dr: Life is an ambiguous term, use sentience to describe a being's value.