I have looked on Scotland's People, FamilySearch and Ancestry but have not been able to find/confirm his birth date
https://www.rootschat.com/forum/index.php?topic=714261.0Could your informant have found it in a family bible, perhaps?
Or was his DoB on his NZ death certificate?
Have you found an death announcement or obituary in a newspaper?
Is his DoB on his gravestone?
I see from the 1861 census that Robert had a sister Agnes, born 1850/1851, and a brother David, born 1852/1853, and note that their baptisms are not in the available records on Scotland's People. The usual reason for a whole family to be missing from records after 1843 is that they were members of the Free Church of Scotland, which split from the Church of Scotland in 1843, or of the Episcopal Church of Scotland, which was separate anyway.
In 1855, at the start of civil registration, the Church of Scotland was required to hand over its registers to the Registrar General for safe keeping, and those registers for the bulk of the registers now digitised and online at Scotland's People, and indexed by FamilySearch, FindMyPast and other web sites.
However the government could not force the other denominations to hand over their records, so the other churches' records are separate. Many surviving registers of free denominations are now held in the National Records of Scotland, and most of these have been digitised and are similarly available on Scotland's People. There is a small number that cannot be digitised (yet) either because it is unclear who actually owns them, or the congregaton they belong to has not agreed to their records being digitised.
The Roman Catholic records have been digitised by agreement with the Roman Catholic Church and are available on Scotland's People and FindMyPast.
It is possible that your lady managed to find, or had a searcher find for her, a stray register which for one reason or another has not been handed over to the National Records or digitised.
I note that the marriage of Robert Terras and Mary Ann Peter Morris was recorded on 31 December 1847. Is it just possible that the actual wedding was held on 28 December 1847, and someone has confused the two dates? You can get the wedding date from the birth certificates of some of the younger children. I recommend going for the birth certificate of the unnamed child born on 18 January 1855 because 1855 birth certificates contain a lot of detail not included in earlier or later records. However the birth certificates of James (1862) and Andrew (1865) should also list the date and place of the parents' wedding.