I still don't understand the Oriental Orthodox (miaphysite) position.
That the Logos died in and because of His flesh.
Christ (one person, two nature) died on the cross, what died? The answer I've seen is Jesus died, but that sounds like avoiding the question.
No, it's not. If anything, you have an improper understanding of what dies. Anhypostatic elements do not die. Simple "whats" do not die. Hypostases die. Death is an experience of a hypostasis. The Incarnate Logos died.
If the person of Jesus Christ died in totality, then his human and diving natures both died?
You are introducing too much division. It would be improper even to say that the flesh of Christ died. Christ died in the flesh is all you can say. So there is no point trying to talk about His divinity dying either.
But if the Jesus died, something died, right?
"The Jesus"? "Something died"? You are speaking too abstractly of the Lord Christ. It was a someone who died.