In the end who's numbering counts?
Eccleston's Doctor considered himself the 9th, as he'd rejected the war Doctor.
Tennant's Doctor likewise called himself the 10th. Both knew there was another regeneration in between, but refused to call him Doctor.
Smith considered himself the 11th until very shortly before his regeneration when he finally claimed and owned his 9th regeneration as Doctor. So Capaldi's Doctor will consider himself the 13th.
And now that he remembers what happened he'll set out to release Gallifrey, whereupon he'll be rewarded with more regenerations. And the internet will explode again.