The victim of impersonation isn't held accountable for the action of the criminal, particularly with banks. That's precisely what identity theft laws protect. I'm in favor of making that process even easier for the victim wherever possible but changing the name is not that.
Someone who steals my passwords can impersonate me, they can not become me. Someone who tricks people into thinking they are me is still not me. An account is not an identity.
My online accounts are not me, and I am not my online accounts.
"1a: the distinguishing character or personality of an individual
2: the condition of being the same with something described or asserted
3a: sameness of essential or generic character in different instances"
Or the Cambridge dictionary:
"a person's name and other facts about who they are:"
Of course, you're always welcome to intentionally pick the incorrect context (going back to Merriam-Webster):
"4: an equation that is satisfied for all values of the symbols"
And just as easily rant the name of the crime has nothing to do with math so it needs to be renamed.
How this is interpreted is the primary problem that needs to be solved in order to legislate the issue, and from the interpretations you're supporting, it seems you're firmly on the side of the parties failing to identify fraudsters correctly, versus the innocent individuals.
As I've stated I'm in favor of the individuals. When doing so, I like to stick to factual and logical reasons why instead of insist the root of the problem is the chosen definition of "identity" does not match my preferred one. The root of the problem is the current process for dealing with identity theft is more burdensome than it needs to be on individuals. If you change the name it's still too burdensome because, outside of you, everyone already interprets "identity" to mean what it should in this case. That the process is bad is not proof the interpretation of the phrase is bad and needs to be fixed, it's proof the process is bad and needs to be fixed.
Just because someone disagrees with one component of your stance does not mean they automatically agree with all opposing positions that followed from it.
I don't think it's possible to avoid mentioning the banks. They are the ones committing the harm against you.
They are a stand-in for numerous other institutions who abuse you. You can take the name "bank" to mean any organization who is defrauded, and then abuses you to obtain repayment for that fraud.
I think it's important to recognize that this is a two-step process. The middle-man in this procedure is crucial, because they are the ones with a lot of power to use the legal system against you. If they were somebody other than a bank or other significant corporation, you'd be able to say, "No, I'm not the John Smith you gave money to. Go away and find that person." The imbalance makes it necessary to define the argument in terms of banks and similar institutions.
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.
This is actually a great example.
If you impersonate me in an e-mail and talk someone into sending a thousand dollars they owed me into some strange account, a reasonable victim isn't going to come to me and say "we're square now, because I paid that fraudster's account what I owed you."
Instead they should admit they made a mistake in assuming it was me based on suspiciously inadequate information and pursue the fraudster if they want their money back.
If you still disagree please try to make an argument without mentioning banks. Identity theft covers a lot more than banking fraud so the explanation shouldn't explain how you want the term to be changed to something which focuses solely on banks.
The processes in place do suck. That has nothing to do with the name of the crime though.