Many transgender people are upset about the IDs in many states having "gender" match their "Birth sex". At the same time your "Birth sex" is still a thing, even if your "gender" is different, and confusion may arise if a person's ID says one gender but they don't fully pass as it, so the person's "birth sex" should also be listed on the ID. Why not list both on driver's licenses and IDs? Have a box that says "birth sex" and a box that says "gender" and have a listing for both. Make it so you can legally change your gender but not your birth sex.
This should also apply to birth certificates. When a birth certificate is changed, the word "gender" should have the person's legally changed gender by it, but another category label "birth sex" should have the sex the person had at birth, whether male, female, or intersexed and that should never change.
Would it not be more simple to have a gender field with "m" for male, "f" for female, and "t" for transgender? I'm not entirely sure that I see why having two different fields is necessary. Essentially, why should the law care what gender you "identify" as? If a person's current gender is not the same as their birth sex, then shouldn't the gender field say "t" to explain that situation?
Only birth sex should be on any legal document. You have the rights to live your life as a man or woman but it doesn't make you a man or woman if it differs from your birth sex and as such should not be indicated in any official legal document.