We'll only know when that gets tested in court, but I'd be willing to bet the answer will be: yes, you have hacked a bank. I find it very hard to believe the justice system would let someone off on some technicality around intention and agents after a serious bank hack.

A web browser can't decide to hack a bank anymore than a LLM can. Neither have any understanding of what a bank is or any will to act on their own. The person who instructs/uses a web browser to hack a bank (even if it's someone else's browser) commits the crime.

> If your web browser hacks a bank, but you didn't know and didn't expect it to, have you hacked a bank?

Depends, as usual. Intent can matter, but depends on the statute (and jurisdiction) in question.