According to the FindMyPast transcription of the 1871, Elizabeth's birthplaces was Boynds, Berwickshire. That's more convincing than Boyude, but it doesn't really get us any further. It also says Stroud in 1881.
So we have the following birthplaces for Elizabeth from the census
1841: Born England
1851: Born Mordington, Berwickshire
1861: Born Berwick, England
1871: Born Boynds, Berwickshire
1881: Born Stroud, Berwickshire
1891: Born England
1901: Born England
There's a place labelled 'Bait's Cross and Strand' on the first edition six-inch map. It's just inside the parish of Berwick, quite close to Mordington Bridge. Is this Straud/Stroud?
https://maps.nls.uk/geo/explore/#zoom=15.7&lat=55.78775&lon=-2.07693&layers=5&b=1The baptisms in Mordington of three siblings of Elizabeth are on Scotland's People
Robert, 13 May 1835
Charles Lillie, 17 July 1837
Agnes, 13 July 1839
Have you looked at those to see if they say where in Mordington the family was living?
Can you post an extract from the original 1871 census so we can see if we can make it out?