As far as I understand it, Gatwa is the Fifteenth with the "second Tennant" Doctor being the Fourteenth (distinct from the Tenth). To add even more fun to this, it's been implied that 15 technically comes from a point in the future of the Doctor’s personal time line ("we do rehab out of order") and 14 will properly regenerate into 15 when he has healed his mental and emotional wounds.
Regarding the "Hurt" Doctor, he 'temporarily' discarded the name of the Doctor during the Time War, reclaiming the name "the Doctor" afterwards (hence Eccleston being the 9th). The official name of the Hurt Doctor is "the War Doctor".
As far as confusion, eh it comes with the territory. At this point I let the BBC sort it out 😊
I'll never forget that Trump supporter complaining that "he's hurting the wrong people" during his administration