2. The Explosion Was Planned, The Reaction Wasn't
It wasn't just the name badge that fans have wrong about the hospital sequence. When Joker destroys Gotham general because why the hell not and theres a delay in the explosions, leading to some serious tigger fumbling, many people are quick to assume that the whole moment wasn't planned.
Not quite, my trivia mistaking friends. Although the delay in the explosion was always intended - there wasn't any !*$% up on the pyrotechnics part - Ledgers reaction certainly wasn't. The thing is, its only the way the Joker reacts to the explosion, drawing attention to the delay, that makes it look like a mistake. Had he just walked off screen (as I assume was originally intended in the script) itd have look like a simple domino-esque explosion.
When you think about it, its a little ridiculous to expect Ledger to react the way he did so spontaneously. If the technical stuff behind the explosions had broken then odds are the take would be useless, so even though he allegedly stayed in character while on set, theres nothing to be gained from a little bit of extended improv.