he would have been christened William !

The entry in the Parish Reg probably read William illegitimate son of ...
The mormons in building the IGI have translated this to William Corteen son of .. whereas until much later in the 19th C (probably with civil registration) he and other children born in similar circumstances would always have been known under their father's name.

Sorry to harp on about this but the IGI is just a database not a 100% accurate transcription of the Parish Reg