I think the concave one is for sharpening DE blades. Maybe the flat side is meant as a barber hone.
You are right on the money, the concave stone is for DE blades, tho I have only seen a few made from actual stone-mostly they come in glass, as glass was a popular honing material for DE blades during the Depression (and before). I also hear that some folks would hone DE blades inside a water glass.I think the concave one is for sharpening DE blades. Maybe the flat side is meant as a barber hone.