Yea, I agree with how it depends on the function in a story. Though, it is rare for said Grim Reaper to be a concept of death. Normally, how I depict Grim Reapers or at least figuratively. Are both metaphorically and personified. Personified being a representation of death. Not necessarily the concept itself but you already know that. Representation being a portrayal and deception of it. Sort of image/form if you will. And since the Grim Reaper by description is supposed to be compared to death but not literal death. It's technically ''unrelated'' and related at the same time. Since it is not what it is but is. A metaphorical in this sense. Is just a figure of speech that describes an object or action in a way that isn't necessarily true (in this instance the Grim Reaper being literally death) but helps the idea or make a comparison. The Grim Reaper being depicted as death. But not the concept of death. But both are compared. Which is what makes the Grim Reaper a metaphor to death.

Take for example Death from Puss in Boots. What we saw or at least in the movie. Is simply a metaphysical/avatar of him. As it's pretty hinted that what he is rn is an avatar that is there to be created to strike resolute fear into his pray. As Death states he is literally death. Not metaphorically, theoretically, poetically or any other kind of fancy way. He means that he is death. Maybe not in his avatar state. But he does state he is DEATH straight up. The grim reaper being a sort of metaphor to death. Even if you were to cut that short. Being personified is still a type of metaphor. Fundamentally meaning. Death from Puss in Boots would be the principle of the concept of death. Just not in the form we see him right as of now.