> If I use a keylogger to grab your passwords and impersonate you in emails, forums, and so on then your identity is stolen

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.