I'll tell you, I don't believe permawhite is the only thing to the Joker. But I think it's up there.
I think your assertation that the the cut smile is just a twist on permawhite is a valid one. I just find that, in my mind, the cut smile, compared to permawhite, is a little mundane. I like the permawhite because it's just so out there. It's a deformity that perfectly matches his personality. Something so weird, and unique, and, at the same time, grim, because it gives him not only the appearance of a clown, but also that of a corpse.
A cut smile seems to be a somewhat watered-down version of that ideal.
I don't mind the cut smile in addition to permawhite. I think that would have been the perfect edge to Nolan's Joker. In fact, I would have really liked it if he was permawhite with the cut smile, and wore black & red makeup to give himself a more clownish appearance. Because, even with the removal of that makeup, he's still every bit as out there.