I guess the sticking point for me on the issue of whether Anthony was Jan Janse/Murad Reis' son is the fact that the marriage certificate issued to Anthony and Grietje Reyniers indicates that he was born in 1607 and Jan Janse was captured and "turned Turk" in 1618. Based on these dates, if Anthony was Jan's son, it would have been by Jan's Dutch wife, yet Anthony was described as mulatto. (I assume that Anthony's description of himself as "a seaman from Cartagena" was just a cover story, but it could explain everything except his name.)
I suppose that it's possible that Anthony just allowed people to assume that he was a famous man's son. His appearance would have backed up the story, and "Jansen" or "Janse" is not an unusual name.