Or you could say that David Tennant is both the 10th and 11th incarnation, as he didnt want to change, so Matt is the 12th and the 13th is yet to be picked.
You could say that, but I wouldn't.
In story reasons would be that the whole hand-absorbing-the-regeneration crap meant that he didn't regenerate, he used the aborted regenerative energy to heal himself instead and the hand absorbed the regeneration. Therefore, he is just the 10th incarnation that nearly -but didn't- regenerate.
In real life reasons, nobody who's casting a new Doctor is going to count that one simple (and lame) sub-sub-sub plot that will probaby be all but forgotten in a year or so except by the very very nerdy like us. If the writer's bible states that the Time Lords have 13 lives (12 regenerations) then they will cast 13 actors and number them thus. They won't bother going back over all the old stories to see if there's an episode where he sort of half-regenerated and try to explain it.
If the show is still going strong when the 13th actor decides to pack it in, then they'll come up with some new explanation at the time, but they won't need to worry about it until that happens.
You won't see any publicity material with David Tennat's face printed twice amongst all the other actors. He is simply the 10th Doctor.