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[/floatr]Perhaps when it's a Japanese organ donor card. The images here are from the English site of the Japan Organ Transport Network. As you can see, the holder must select from three options: a list of organs available if brain death is declared; a list of organs in the event cardiac death is declared, lastly, no organs. Organ donation is still in its infancy in Japan. In particular, there are concerns about whether brain death is a sufficient condition for organs to be removed. It's possible that the third option is a legal requirement, or some other safeguard, to confirm a donor's true intention. You would expect cardholders to select one or both of the first two options. However, this Yomiuri article says differently:
The percentage of people refusing to donate their organs if they are declared brain-dead increased from 2 percent to 10 percent in August, among those who newly registered with the Japan Organ Transplant Network Web site. On Aug. 9, in the first case approved under the revised Organ Transplant Law, organs were harvested from a brain-dead patient with the consent of the patient's family, but without confirming the intent of the patient. The sharp increase in network registrants who chose not to become organ donors likely was triggered by the incident... Until August, only 2 percent of registrants chose the third category. However, after Aug. 9, the number of overall registrants grew while the percentage of those choosing not to become organ donors jumped to 10 percent.
As far as I can understand - and I may be mistaken - some people are actively registering with an organ donor association to get a card which says they will not donate organs under any circumstances, not simply that they don't want to do so when brain dead. Presumably, they are concerned not having anything which expresses their preference might leave them open to some misunderstanding should the worst happen. Personally, I'm inclined to think whatever slim chance of confusion there might be would more likely occur if I was carrying any kind of donor card, regardless of which option I'd selected.
I don't know about organ donation schemes in countries other than Britain, where we have no such third option on a donor card. Perhaps it's a common feature elsewhere. I can't help feeling a pang of sympathy for a medical professional who is briefly elated to find the card only to flip it over and find out nothing is available.

