I am going to preference my post saying I have exerted conscious effort towards making my worlds live and breathe ever since highschool, so getting on 30 years now. Before the term was ever coined, I suppose. I'm not a fan of nerd jargon. So I tend to think of things in terms of my understanding of their plain English meaning.
That said, the world going on whether the PCs do or do not engage? At one time that was a consideration in what I did. And sometimes it still is. I just don't see it as a vital characteristic to creating a living world. I ask the following question: If the stuff that happens in the background does not affect the PCs at all, if they do not engage it, if the players never experience it, does it even matter?
To the players? Not at all. To me? It might. So the relative importance of this characteristic seems highly dependent upon how I balance out whether you're I'm creating the living world for my players or myself. It might be one, the other, or a little of both. But how I balance this out seems to be a side consideration separate from the question of what is a living world.
For me, the defining characteristic is does the world conceptually behave as a living creature would. Consider the following questions. For it to be a "living world" I think the answer to the first two has to be "yes"--the last two are sort of extra credit. Think of it as stretch goals. Or advanced living world design.
Does it react and respond to stimulus (player action)?
Does it have have a "personality" of its own that persists in spite of what the players do?
Does the world have a "will" its own, such that its evolution is not strictly deterministic?
Is the world capable of having a "soul-changing" experience, where the unfolding of extraordinary events can alter the world's underlying "code"?
One definition I've encountered to the term classifies "living world" as being a type of "sandbox" where "sandbox" is defined as being open to destruction of the premise. That seems to imply a strong "Yes" to question #1 but seems to fail question #2. And so it doesn't quite jive with what a Living World is to me.