Criminal identity theft is one class of examples. An example scenario from this class is someone passes your identifiers off as theirs while committing a crime. Nobody was defrauded, no money exchanged, but you'll still want to report identity theft.

Claiming identity theft is precisely the process to notify the bank (or others) they cannot legally abuse you to get repayment for that fraud or you are not responsible for those crimes or whatever occured on your behalf. Under identity theft laws they are responsible for the loss due to fraud, not you. The same as credit card companies. The legal system is used but as much by you saying "I didn't buy that house, clear my records and eat the losses" as by the bank initially saying "this person didn't pay their loan". To not involve the legal system by both parties just opens up an even worse can of worms of fraud.

One thing I do agree on is that anything that can reasonably be done to make the process easier on the victim of identity theft should be as the process is too hard on them right now. Probably more fines to most middlemen to increase the cost further beyond their losses. I just don't think changing the name of the crime has anything to do with that kind of improvement.