The 5e rules use the natural language philosophy. How the words are used naturally holds sway, so if you went to a stranger on the street and asked this question, what would the answer be? They'd say that you cannot be living and dead at the same time, and if they knew about undead, they'd say that undead was neither living, nor dead. That means that they are all mutually exclusive by how 5e works.
You could of course make it different for your game.
You'd have to make the call as DM. Rulings over rules.
I've been playing D&D since 1983 and I've never seen or heard of anyone who didn't treat an undead as dead after it was "killed."