The guy called him the 13th! Stop splitting hairs! You're like a grammar nazi, but this has nothing to do with grammar!
I'm not splitting hairs, the simple fact is that the names First Doctor, Second Doctor, Third Docter, etc.. are names that have always been used out of universe. The character, in show, has always just called himself the Doctor with no numbering distinction between his different incarnations. In the end it's up to BBC, and the names they use on their merchandising, to decide what the different incarnations are called. It seems that they have and Tennant is still the Tenth Doctor, Smith is still the Eleventh Doctor and Hurt is the War Doctor. Most fans will probably follow suit due to it be easier not to use names different from what is on the DVDs, t-shirts, etc.. that they own.