I have had a Eureka moment (may be an Australian saying), but have found the one piece of the jigsaw that joins the Elizabeth Davies giving birth to the illegitimate William Davies to the Edward and Margaret Davies that had a grandson named William Davies staying with them in 1881. Previously, it was ASSUMED that these two events were linked.
A search of the birth/baptism records revealed an entry when I used Eglwys Rhos as the location. It returned a Baptism entry listing a William Davies having a mother 'Elizth'. Looking at the image of this record reveal that it lists the birth as Illegitimate and most importantly the Abode as "LG Bricks" which would seem to match with 1871 Census address of "Brickfield" for Edward and Margaret Davies.
Thank you, thank you, all for helping me to arrive at this point.