To dispel one inaccuracy that appears to be in your question (please correct me if I'm wrong about this): the descendents of Castamir were not the Black Númenoreans.
Black Númenoreans come far earlier, from the time of Sauron's captivity in Númenor until his overthrow at the end of the Second Age, and a footnote to Return of the King Appendix A gives their history:
The great cape and land-locked firth of Umbar had been Númenorean land since days of old; but it was a stronghold of the King's Men, who were afterwards called the Black Númenoreans, corrupted by Sauron, and who hated above all the followers of Elendil. After the fall of Sauron their race swiftly dwindled or became merged with the Men of Middle-earth, but they inherited without lessening their hatred of Gondor.
This doesn't say who it was called them the Black Númenoreans but it seems clear that it was a given, rather than chosen, name.
It also mentions that they were the King's Men of Númenor, and this is defined in the Akallabeth as follows:
Then Tar-Ancalimon, son of Atanamir, became King, and he was of like mind; and in his day the people of Nœmenor became divided. On the one hand was the greater party, and they were called the Kings Men, and they grew proud and were estranged from the Eldar and the Valar.
It therefore seems most likely (although it's nowhere confirmed as far as I'm aware) that Black Númenoreans would have self-identified as "King's Men".
The descendents of Castamir considered themselves ethnically purer, because the line of the kings of Gondor had become intermingled with lesser Men of Rhovanion in the time of Eldacar, thus precipitating the Kin-strife, as we read in Appendix A:
After the return of Eldacar the blood of the kingly house and other houses of the Dunedain became more mingled with that of lesser Men. For many of the great had been slain in the Kin-strife; while Eldacar showed favour to the Northmen, by whose help he had regained the crown, and the people of Gondor were replenished by great numbers that came from Rhovanion.
Since the King's Men were also (in origin) ethnically pure Númenoreans (although that didn't last long), the descendents of Castamir would have that much in common with them.