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.
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.