Personally, I don't think the issue is in the appearance of either "bleached" skin or disfiguring scars - that it lies within the resulting psychosis as a result of either state's birth.
The traditional origin
When he was bleached, his state of mind cracked after seeing himself in the mirror - what had become of him. He saw what any normal person would see: a freak. The only difference, however, is that something suddenly clicked. A light bulb went on inside of his head and he saw the world through new eyes. Not only did he see
himself as beautiful, but the chaos of the world - the big Joke.
In this case, it was realizing the results of the disfiguring event that caused his mind to crack - or, from his p.o.v., caused his eyes to open.
"Unclear" TDK origin
From what we can gather, this new Joker has two disfiguring scars: a clean, upward-curling cut in his right cheek, and a messy, "bubbly-looking" scar on his left. Judging from this, we can assume that one of the scars was given to him by outside forces.
It is at this point that he may have changed; become twisted. Like the traditional origin, it seems to have taken a traumatic event to cause him to crack. But, unlike that origin, it may have been simply the event that did this and not seeing what he looks like in a reflection.
Now, this doesn't necessarily mean that he's going to start calling himself The Joker just yet, but that his mind has changed in a similar fashion. Though, upon seeing his reflection some time after this - as he most likely had done - rather than detest this freak looking back at him, he embraces it. In fact, he seems to have made it even better (in his eyes) by completing that permanent smile etched into the flesh on his face. He sees a clown smiling back at him, and he must have be laughing at the Joke at this point...
But it doesn't end there - he completes the image by not only enhancing his "perfect smile" by smearing red along its length, but by painting this "enhancement" upon a white canvas in order, of course, to make it as vibrant as possible.
And what's a clown without something around the eyes? What stands out better than anything else against white? Hwy, black, of course
- - - - - - - -
To me, the core of the character is in his descent (or ascent

) into insanity and his view of both himself and the world around him. Because of this, the Joker in TDK shares with his comic-book counterpart the most basic essence of the Joker.
And the essential aspect of a character, in my honest opinion, is the most important thing to get right.

