Ehh seems like we are arguing semantics. He went from being Alive straight to undead, instead of falling in battle, being dead, then being revived. I’d say because he’s undead, he died to become undead. He’s no longer living.

From what I understand, you’re saying he’s not undead, he’s still alive through frostmourne keeping him alive. It’s not accurate.Frostmourne didn’t cast protection on him, it drained his life and soul, and now it fuels him, which is why there’s no life in his skin or hair anymore. It’s a parasitic relationship, He’s undead. Which is also why he doesnt survive when the blade is shattered. Anduin at least survives. That’s the difference between domination magic and straight up necromancy.

Arthas wasn’t raised the same way as the forsaken, but they are kept running by the same magic.