This puzzle may relate to one of the many schisms in Scottish Presbyterian Church but I'm hoping someone with local knowledge might be able to explain.
Both details from OPRs on Scotland's People.
Church of Scotland , Eyemouth , 1847
" Jane, daughter of Robert Bryce, paper maker and of Agnes McGechan his wife, was born on the seventh day of June Eighteen Hundred and Forty Seven years and was baptized by the Rev Mr Pearson Eyemouth"
Next entry
"Robert , son of Robert Bryce, paper maker , and of Agnes McGechan his wife , was born on the twentieth day of March Eighteen Hundred and Fifty One years and was baptized by the Rev Mr Pearson , Eyemouth
ALSO in the register for Eyemouth United Secession Church, 1847
"Robert and Agnes Bryce child born June 7th 1847 and baptized Jane by the Rev Thomas Pearson July 25th 1847"
The only other known baptism for a child of Robert and Agnes is in Ayton OPR for :-
"James, lawful son of Robert Bryce Papermaker , residing at Millbank and Agnes McGechan his Spouse was born on Oct 20th 1844 and baptised by the Rev Mr Ure Minister in Ayton"
From the 1851 census , James is the eldest and there is a further sibling Alexander, born circa 1850
Robert and Agnes and family are living with her parents at Eyemouth Mill.
I have been unable to locate a marriage record for Robert and Agnes. Robert was born c1816 in Kilpatrick , Dumbartonshire.
James married Marion Wright in Ayton 24/11/1865 and had ten children, Agnes, Jane Burglass, Elizabeth , Robert Alexander, James - married Jane Tait 22/4/1897 with son James 28/1/1909 , John, William , Marian, Grace Bone , and Peter.
I am bemused by the double baptism entry for Jane Bryce.