After posting my essentially guesswork-based opinion on what criteria should be used for determining a person's sex, I went Googling for information from people who actually have studied this. I found a good report (pdf) from the Canadian Academy of Sport Medicine which gave a rundown of the history of testing in international track and field. It discusses the attempt to determine whether the athlete's "gender"offered any competitive advantage.
This emphasis on practical consequences stuck me as exactly right, and caused me to reconsider my original statement that DNA testing would be the "better" way to go. If you've taken on all the physical characteristics of a woman and are living as a woman, accepting all the advantages and disadvantages of femininity,then how you were born is no longer of any relevance. If I lose a leg or my sight, or my hair color goes gray, the fact that I was born differently doesn't matter, nor does whether or not it was voluntary like a sex change. I'm still living with my current status, and it should be considered as part of my legal situation.
Mind you, this would screw over the couple in the New York Times article. I would suggest they draw up the appropriate legal documents to make sure inheritance and other issues are handled according to their wishes.
I still ponder the sexual-preference consequences, particularly considering the idea that sexual preference is genetic. Loving a person enough to go against your personal sexual preference in order to have sex with that person would be a decision few people could make. Then imagine explaining it: "Well, she was a man when we married, but I love her enough to have lesbian sex now that she has become a woman."