I would say.. Yeah, Once the names been changed legally, you would need to write that new name in the deathnote in order to kill someone.
Though... your birth name could still be used against you. Since you can't exactly change your borth certificates details, because you can only be born once. So by changing your name, your just giving someone anouther name to be able to use to kill you. Since both names would be legal, in a way.
Or maybe not. I don't know lol. I'd need to know more about the laws regarding changing your name legally. Laws might even vary depending on your country. Plus theres the dethnotes rules to consider aswel, and no one but the maker of the Death note can make the books rules lol
It's what the person considers to be their name; the birth certificate has nothing to do with it.
For example: If a very young girl was found on the street (and wasn't aware of her name) and was taken in by a family who called her "Sue", and yet on her birth certificate she was named "Lindsay", Sue would be the name written in the Death Note.
So, returning to the original question, it is most likely their new name would be the one that would be written in the Death Note if they considered it to be THEM, if that makes any sense. If it was simply to avoid the power of the Death Note that they changed their name, though, then it's basically the same as an alias and won't work in the Death Note.
It is not a matter of legal issues, but personal belief/attachment to the name.