The Doctor has had 13 incarnations. Hurt was the 9th. He regenerated from McGann, and regenerated into Eccleston.
That makes him the 9th Doctor by every definition of the word.
That's fine, you have a perfect right to call him that. My problem is that you seem to be saying that I'm wrong to want to go along with what the BBC has set up as the official names. I feel I have a perfect right to choose to do that even though you seem to think I don't.
Eccleston is still officially the Ninth Doctor, Tennant is still officially the Tenth Doctor and Smith is still officially the Eleventh Doctor. Chronologically in the real world Hurt is actually the twelveth Doctor, but is called the War Doctor instead. Will Capaldi be called officially the Thirteenth Doctor? Of that I have no doubt, but that's not going to change what Eccleston to Smith to Hurt are officially.