to take points one by one: I had already seen an entry in old IGI (now it seems replaced by new site) that had a Patrick Cain had an illeg child - however patron submitted so after checking I seem to have marked this in my database as questionable - however since that time I have a data base of entries found in Presentments (these cover many topics not just ch born out of wedlock) so I thought I'd look there and found and later confirmed the entries - Lois appears to have found on the new familysearch site records of the bapt (I will check these on Monday) - the presentments would not be available to anyone off Island (tho some are included with wills for some years)
2nd point - I am not close to your researcher - I see her sometimes in Museum but no more than that.
3rd of course the children's names will match between my reading of Ann Caine als Corlet widow of John Caine Miller and your reading as putative wife of Adam except I can acount for all without introducing foster children etc as well as the absence of the husb's name - my reading also connects with lib vas, the deeds + the burial registers of entries for some of named children
4th - as I have repeated time and time agin there were 2 Adam Caines - one in Michael (farm near Orrisdale) and Adam in Knockan Ballaugh - he was the father of the twins - his wife was Ann Callister (took some time to find name but she is named in full in one of the later deeds) - she d 1739 'wife of Adam' buried in Ballaugh - Adam was still alive and named in will - his buial 'of the Knockan' is noted in Ballaugh Jan 1768 but no will (he had already settled property on his son Patrick
Patrk m Ellinor Clarke - she d 1752 - there are property transfers + the mortgage Dated 5 Feb 1744; Pat Cain with consent of wife Ellinor Cain for consideration of £8 mortgage to friend Phillip Crain, taylor. Ballaugh a certain part of the qtrland of Knockan vizt the flatt called Cain's Flatt ;9 Jan 1764 - Philip Crain acks paid by Pat Cain - Pat appears to have remarried shortly afterwards (necessary as he had young children) to Isabel Kelly