https://familysearch.org/ark:/61903/2:1:M6GN-JBQ I gather this is your source for the ‘birth info from Family Search’. It is of course from a submitted source, and so it needs to be validated against your own research.
So do you have the actual information from the birth registration for Mary May COOPER?
Here is the information that is gleaned from the ONLINE INDEX freely available at the following link
Mary May, daughter of August and Mary COOPER, registered in the Lismore District of NSW, with a birth date of 12 May 1880. Here is the reference no. for the registration of the birth of Mary May COOPER: #23676
http://www.bdm.nsw.gov.au/Pages/family-history/family-history.aspx The actual official transcription of the registration should give you responses to the following headings:
Date and place of birth of child :
Name and whether present or not:Sex:Father’s name (1), occupation (2), age (3) and birthplace (4):Date and place of marriage (1) – previous issue (2):Mother’s name and maiden surname (1) , age (2) and birthplace (3):Informant (JM notes this is recorded as their full name, relationship to the baby being registered, and usual address. Not all informants in the 1880s were literate, and the registration system was based on a verbal exchange between the recording clerk/deputy registrar and the informant, often in rural areas across a high and wide counter within a district court house. )
Witness (1), (2), (3): (JM notes that option (1) is usually for the Doctor, option (2) is usually for the Midwife and option (3) is usually for the lay person present to assist, so often in the late 19th Century in rural NSW that lay person would be the baby’s maternal relative (so the baby’s mum’s mum/aunt/sister/cousin).
Particulars of registration: (District registrar, Date, Location)
Name if added after Registration of birth:I note that at NSW BDM online index there’s a death registration for Augustus C COOPER in 1892, registered in the Tweed River district. The NSW BDM's online index displays given names of his parents as recorded on that registration. That NSW BDM district is next to the NSW BDM Lismore district. If that is Mary's dad, Mary would have been 12 years of age when her father died. If her mother re-married in NSW then I would suggest you also consider seeking out that later marriage certificate to confirm the details about Mary’s maternal grandparents as that information would be sought by the officiating minister/registrar.
May I express concern about some conflicting info. Your topic heading reads:
"
augustus clare COOPER[ B1856 NSW] -m-Mary MCDONOUGH[ B1858] in 1878 NSW."
but you also write
...... If you like to look, my heading states birthdates and marriage of Mary May's parents and Augustus is not listed as being born in NSW
So, from your research, .... was Augustus born in NSW or not.....
Add .... You see there's this baptism
I suspect that your Augustus Clare was born 13 April 1854, Bullinah, NSW (perhaps that would be today’s Ballina NSW), and baptised 17 August 1854, and that his mum was Augusta CLARE.
https://familysearch.org/ark:/61903/1:1:XTHK-C3Thttps://familysearch.org/ark:/61903/1:1:XTCZ-LZN http://www.bullinahahs.org.au/ It is also listed at NSW BDM online index
Cheers, JM