Hi Jamesdon,
I searched the records I have and this is what I found. All dates are of burial. The plot records only begin about 1878 but the gravediggers day books go back to 1861, you will see that there is different information in each type.
You are right about the Barr family - millions of them

The owners of the Barr plot, in Victorian times were, as you'd expect, David Barr & Agnes Dalziel.
There are 13 recorded burials from 1879 but there would probably have been at least double that amount before then. The plot is marked "ground closed" after the 1918 burial, this was because the law forbid the removal of old bones by the gravedigger to make room for new burials. So, in this plot are,
In the South lair -
AGNES BARR, aged 29 years, 19th November 1884.
DAVID BARR, aged 10 years & 6 months, 6th July 1900.
JOHN BARR, aged 7 years, 6th July 1900.
MARGARET BARR, aged 9 weeks, 8th September 1896.
In the Centre lair -
DAVID BARR, aged 17 years, 14th February 1890.
AGNES BARR, aged 89 years, 28th November 1895.
ISABELLA BARR, aged 34 years, 6th October 1897.
CATHERINE STEWART, (no age given, probably an infant) 30th October 1918.
In the North lair -
AGNES BARR, aged 17 years, 31st March 1879.
JANET BARR, aged 2 years & 6 months, 7th June 1879.
AGNES BARR, aged 2 years, 30th November 1880.
DAVID McLEAN, aged 2 years & 6 months, 14th November 1906.
JESSIE BARR, aged 68 years, 8th March 1915.
Ground closed.
From the gravediggers day books -
JESSIE BARR, housewife, Wishaw, aged 68 years, married.
Parents - Alexander Bell & Elizabeth Jardine.
Buried in the old churchyard on 8th March 1915.
DAVID B. McLEAN, Morningside, aged 2 years & 6 months.
Parents - William McLean & Janet Barr.
Old churchyard on 14th November 1906.
JOHN BARR, watchman, Wishaw, aged 55 years, married.
Parents - David Barr & Agnes Dalziel.
Old churchyard on 4th June 1878.
AGNES BAILLIE, housewife, Wishaw, aged 31 years, married.
Parents - David Barr & Agnes Dalziel.
Old churchyard on 22nd November 1869.
MARION BAILLIE, Wishaw, aged 5 years.
Parents - James Baillie & Agnes Barr.
Old churchyard on 24th December 1868.
JAMES BAILLIE, Wishaw, aged 6 years.
Parents - James Baillie & Agnes Barr.
Old churchyard on 9th May 1865.
MARGARET BAILLIE, Wishaw, aged 7 weeks.
Parents - James Baillie & Agnes Barr.
Old churchyard on 24th February 1865.
Unless I can find the death and parents of James Baillie, husband of Agnes Barr, there is no way of knowing if this family is buried in the Barr plot.
Not so many Lightbodys!
The plot owner, in Victorian times, was Thomas Lightbody.
In the South lair -
EUPHEMIA LIGHTBODY, aged 84 years, 19th July 1880.
W. McLEAN MOFFAT, 14 years, 16th November 1889.
In the North lair -
THOMAS BLACK, aged 69 years, 28th December 1901.
From the gravediggers day books -
JANET LIGHTBODY, housekeeper, Carluke, aged 60 years.
Parents - Thomas Lightbody & Agnes Yeatts.
Old churchyard on 3rd March 1868.
ANN YOUNG, Wishaw, aged 69 years, widow.
Parents - Thomas Lightbody & Agnes Yates.
Old churchyard on 27th November 1871.
EUPHEMIA LIGHTBODY, servant, Carluke, aged 84 years, unmarried.
Parents - Thomas Lightbody & Agnes Yett.
Old churchyard on 19th July 1880.
I have extracted this Lightbody information before on Rootschat, perhaps Monica or some other knowledgable member can find the link and add it to this post? Looks like someone else is researching this family.
Thats all I can give you at the moment, I am in Paris until the 16th (pause to let Monica roll her eyes)
If you send me names & spouses names for the other children of Thomas & Agnes I will have another search for them.