LS
Lou Scannon
Nobody ever gives War Doctor a number.
Although he is generally credited as the "War Doctor" (for simplicity's sake), in the context of the narrative he isn't a "Doctor" at all...
For certain reasons within the fictional narrative, the John Hurt incarnation renounces the adoptive moniker of "Doctor" (and the pacifistic promise-to-himself which it is there to remind himself of - as explained in some Matt Smith-era dialogue/blurb). I suppose that *technically* this decision is made (arguably under duress?) by the Paul McGann incarnation in his dying moments, during that online mini-episode prequel to the 50th anniversary special.
Therefore, although Hurt is the true 9th incarnation of the character, he isn't a "Doctor". Hence, the "incarnation numbers" and "Doctor numbers" then diverge - e.g. Christopher Eccleston is the 10th incarnation but only the 9th "Doctor", and so on.
As others have already said, the "wasted" regeneration that happened mid-Tennant era means that Tennant is effectively both the 11th & 12th incarnations (and yet only the 10th "Doctor" throughout), and so the two different numbering systems diverge still further.
Things will now always remain a bit complex, numbering-system wise - e.g. Capaldi being simultaneously all of the following things: The 12th "Doctor" / the 14th overall incarnation / the 1st incarnation of the character's "2nd life cycle" (as granted to the aged/dying Matt Smith incarnation by the Time Lords).
Who said science-fiction is complicated?!