Elizabeth Miller

There is a bit of a mystery about Elizabeth - her surname.  Jane Cooper's marriage gives her mother as Elizabeth Miller but unfortunately no birth record can be found for Jane, although it would probably only give Elizabeth's first name anyway as it was in the days of the Parish Records.  Neither can a marriage record be found for Elizabeth.

However, a death record is available and, confirming that it is her by mentioning that she was married to James Cooper, Farm Labourer, it gives her name as Elizabeth Stobo Cooper with Henry Miller, Veterinary Surgeon as her father.  From this we can deduce that she had been called Elizabeth Stobo Miller but that middle name is significant.  She was born, a twin to Robert Miller, as Elisabeth Stobie Miller, confirming the middle name though correcting the spelling.  Now comes the confusing part.

Only one child was born to James Cooper and Elizabeth Miller and five were born to James Cooper and Elizabeth Stobie.  All the names match the conventional patterns for assumed parents so, all things considered, Elizabeth Miller is Elizabeth Stobie.  What I can't understand is why she would drop the Miller.  Her twin brother didn't have Stobie as a middle name so it isn't an alternative surname, it's just a middle name.  Did she fall out with her father, a vet, by marrying below her level, to an illiterate ploughman (he signed with an x) and decided to lose the link to the family?  Indeed, I can't find a marriage so maybe it was just common law although even irregular marriages were usually recorded.  You are allowed to call yourself what you want in Scotland so she wouldn't be doing anything wrong if she did change her name.  I'd be more confident in this theory if it was the first child who was born to Miller and the subsequent ones to Stobie but it was the third.  I'm open to suggestions to any other theories.