As far as I'm concerned he never became Two-Face. Two-Face is a comic book gimmicky character that makes the assholes in the academy cringe because they've forgotten how to be young, and has a split style suit and commits crimes based on twos. What we saw in the film was Harvey Dent with an axe to grind and a flesh-wound.
He's like "yo give me yo insurance information!" and I'm like "I didn't hit you with no car, I hit you with this mack truck I call a fist!"