It seems unlikely to me that what your legal name is, is what matters. If it did, that would mean you couldn't kill anyone who is not legally documented anywhere using the Death Note, which seems like an odd restriction for a supernatural notebook to have (there's also the matter of people who are documented under different names in different jurisdictions or with different spellings etc.).

So I imagine it's about what either the writer or the writee acknowledges as their name. However, one of the rules of the Death Note is:

This note will not take effect unless the writer has the person's face in their mind when writing his/her name. Therefore, people sharing the same name will not be affected.

This suggests to me that it's the writer's cognition that matters, meaning this wouldn't work because the writer would still acknowledge someone's dead name as their actual name.