The article says that they have no doubt that he was born in Hawaii. They can say that if the record on file actually says that he was born in Hawaii. If they said "there's no doubt he's a citizen" THEN I would agree with you that the laws in Hawaii would allow for doubt. If they had merely said that they had a birth certificate on file, then too, I would agree with you that it doesn't prove citizenship. However, if they have a bc on file, then someone could have looked at it and seen the place of birth and made such a statement.
Of course, now we get into, "Did someone actually look at the place of birth or did they just assume since there was a certificate, he was born in Hawaii?" Well, I assume that Hawaii Health Dept officials know Hawaii law better than most as it pertains to this sort of thing, so they would know that a Hawaii BC could be issued w/o actually having been born in Hawaii, so they would know to look at such a thing before declaring they have no doubts, especially since no one is really disputing that there is a bc on file in Hawaii, the only dispute is what it actually says regarding birthplace.
There is also the credibility of the source and if someone who has access to viewing BCs actually made such a statement. AP would be likely to confirm the credibility of the source, but you never know. :shrug: