To be honest, that part always confused me. But I don't know if its lousy game mechanics or an enchantment (considering that actually saying that you're the Gray Fox, following The Ultimate Heist, can be comprehended). But still, the cowl doesn't erase the wearer's name from history anymore, just masks it.
Think of it this way, the Grey Fox is always the Grey Fox, no matter who wears it. However, when one wears the Cowl, people see the Grey Fox, not the person under the cowl. When it's off, the Fox disappears.
Don't think too much of it, it's Nocturnal's way of screwing with people. It is her sphere after all.