The War Doctor wasn't the Doctor because he abandoned the promise he made and the name along with it. So, Tennant was the 10th. He didn't fully regenerate that other time in the Meta-Crisis, so he was only/still 10. The Other Tennant is a human, not a Time Lord, and so not the Doctor. Plus, he lives in another universe. The Doctor's Daughter was also a clone and also not a new Doctor, because she wasn't the product of a regeneration.
Then we had Smith, Capaldi and Whittaker as 11, 12 and 13.
13 regenerated into 14 (Tennant again). Then the Bi-generation happened, which again wasn't a regeneration from 14's POV (same face, same personality), but was a full regeneration in that it produced a new Doctor, number 15.
So, they're only a new Doctor if they are a product of a full regeneration, who uses the name 'the Doctor', isn't a clone, is a Time Lord, and isn't a weird spin-off (I'm looking at you, Peter Cushing).
I'm not saying the above makes 'sense' outside of the Whoniverse, but I think it's a reasonably accurate summary of the official position!