Gill,
I'm not sure what the position is for name indexes for Lancashire for 1861 and 1871 - but you never know if you might just get lucky with a more precise birthplace in 1861 or 1871 ... It does sometimes happen!
I take it that James is the James on FamilySearch as son of John MOYES and Margaret, b 14 Jan, bap 12 Feb 1865, St Peter, Liverpool (I guess a look at the film of the parish register would be worth while just in case there is any clue there - witnesses names?). Is Catherine Susan, same parents' names, bap 21 Aug 1867, St-James-the-Less, Liverpool, not yours? She is presumably the Catherine Susan MOYES on FreeBMD who d Sep qtr 1867 age 0. I note that an Alexander and a Margaret MOYES (no ages given) died Mar qtr 1864 in Liverpool.
If John and Margaret were married in 1859, it does seem likely that there would have been children before James in 1865... There are no MOYES births on FreeBMD in Liverpool during those years; have you tried 1837online??
Where was John in 1881? I take it that the relevant entry is Margaret MOYES, M, 44, b Scotland, Dressmaker and James, son, 16, b Lancashire, India Rubber Worker. That lists Margaret as Married (i.e. not Widowed). If there were older children (born between 1859 and 1865) they could well be working away from home.
Well, if no solutions are forthcoming from the information you already have or can get, let's hope that the 1841 and 1851 Scotland censuses when indexed and released will solve your puzzle.
Good luck,
JAP