It's actually a little weirder than that! When injured or killed, all of the character's parts rearrange themselves back to the shape they had when the character was first made immortal. A one-off line in the source novels implies it's a result of some otherworldly eldritch hivemind they infused themselves with via an alchemical elixir. (Anime example here on youtube-- sorry for the lack of subs!)

Would the same circumstances apply re: lethal harm versus minor injuries? Or would it be treated differently from regeneration?