There are secondhand references to Mother Carey's appearance in old rhyme books, but I haven't been able find them.
The Cook and "Historical Account" (Jim Dixon, above) and the reference below suggest that there was a prior use of Mother Carey.
Carteret, 1767, in Hawkesworth's Voyages (1773): "The petrels, to whom the sailors have given the name of Mother Carey's chickens..."
Marryat (1836) and others spelled it 'Cary'. OED
Where did they get the name Mother Carey?

In 1864 an article in Athenaeum, 558/2, noted that Mother Cary's chickens was a sailors name for snow, and that the 'Mater cara' of Levantine sailors, but the OED entry gives no further data, so I dunno.