My INGLIS relatives were from Haddington, East Lothian. I did find 14 Inglis marriages in Cambusnethan in the last 1860s including two for a James James INGLIS m 23 Nov 1860 to Rachel MARTIN James INGLIS m 31 Dec 1867 to Frances MURRAY
There were only four children born to a James INGLIS in Cambusnethan around that time: Christina Hamilton b 7 Jan 1861 dau James INGLIS / Rachel STOBO MARTIN Elizabeth b 21 Jan 1865 dau James INGLIS / Ann GRAY Ann b 4 Jun 1865 dau James INGLIS / Ann GRAY Janet b 5 Oct 1868 dau James INGLIS / Frances MURRAY
The IGI has a James INGLIS born to John INGLIS and Elizabeth LINDSAY on 16 Oct 1830 in Wishawtown, Cambusnethan
Don't know if any of these are related but wish you the best in your search.