Both are true in a sense. Peter Capaldi is the 14th body of the Time Lord who usually calls himself the Doctor, the 13th incarnation, who has regenerated 13 times, and the 12th to call himself the Doctor.
The Doctor has had 14 bodies; one he was born with, and 13 more created via regeneration. The First Doctor was the first body and incarnation of the Doctor, the Second Doctor was the second, up to the Eighth Doctor being the eighth incarnation and body. Then, with his 8th regeneration, the Eighth Doctor turned into the War Doctor, who did not call himself "the Doctor" out of shame ("The Day of the Doctor"). The Ninth Doctor was the 10th incarnation and reclaimed the title of "Doctor". He then regenerated for a 10th time and became the Tenth Doctor, which was the Doctor's 11th body. Then, in "Journey's End", the Tenth Doctor began to regenerate for the 11th time, but funnelled the extra energy after basic healing into his spare hand, so a regeneration was used up but the Doctor did not in fact change incarnations. However, that hand then grew into its own body, the Meta-Crisis Tenth Doctor, and can be considered the 12th body of the Doctor, but not one of his incarnations. Regular Ten then regenerated into Eleven, who was then granted a new cycle of regenerations, since he'd by now used all 12 he was originally given, and so regenerated for the 13th time into the Twelfth Doctor, who is the Doctor's 14th body and 13th incarnation.
In short, once Capaldi plays the Doctor, the Doctor has had 14 bodies including the War Doctor and the Meta-Crisis Tenth Doctor, 13 incarnations excluding the Meta-Crisis Tenth Doctor (since he was a separate being), which used 13 regenerations. However, there are only 12 Doctors as the War Doctor and the Meta-Crisis Tenth Doctor don't count as "the Doctor", since a) the War Doctor didn't call himself the Doctor and b) the Meta-Crisis Tenth Doctor was a separate body to the main Doctor.
From a production/branding perspective, the BBC has officially declared him the Twelfth Doctor, because as explained in "The Time of the Doctor", the War Doctor did not count in the official numbering, nor did the aborted Tenth Doctor regeneration. Titan Comics, North American licence holder for producing original Doctor Who comic books, also officially refers to him as the Twelfth Doctor.