well you did trouble me further, I had spent a further hour searching for your query and was about to give up when I thought of spelling variants and found Francis Conner which I then placed near the top of my reply where Francis Connor was noted
so my further notes :-
Carnmoney area of greater Belfast
Connor family is living in 1901 at 13 in Whitewell (part of) (Carnmoney, with an Owen Carroll from Co Louth who might be Mary Ann’s brother
census1911 This Frances Connor is listed under birth 1904 in familysearch
51 in Collinward (Carnmoney, Antrim)
Connor Henry 45 Male Head of Family Church of Ireland County Antrim Labourer Brick Field Cannot read or write {Yet he signs but the signature has a space after C his wifes name is spelt the same so strongly suggests signed by her see
http://www.census.nationalarchives.ie/reels/nai001540433/ }
Connor Mary Ann 44 Female Wife Roman Catholic County Louth - Read and write - Married - 22 6 6
children all six
Connor William John 21 Male Son Roman Catholic County Antrim Labourer Farm Read and write - Single - - - -
Connor Thomas Henry 18 Male Son Roman Catholic Co Antrim Milk Server Read and write - Single - - - -
Connor Patrick Joseph 16 Male Son Roman Catholic Co Antrim Milk Server Read and write -
Connor Maggie 14 Female Daughter Roman Catholic Co Antrim Mill Worker Spinning Mill Read and write
Connor Mary Alice 11 Female Daughter Roman Catholic Co Antrim Scholar Read and write
Connor Frances 7 Male Son Roman Catholic Co Antrim Scholar Read and write - Single - - - -
I cannot connect this Frances to Balywiilin
BALLYWILLIN CIVIL PARISH from Familysearch I learnt Ballywillin, or Milltown, is a parish in the Poor Law Union of Coleraine, partly in the barony of Lower Dunluce, county Antrim, but mostly in the North-East liberties of Coleraine, county Londonderry. Portrush is also in the Ballywillin civil parish as is Francis Conner in Ballywillin townland.
in the 1901 census there is one Conner and two Connor families in Nos 2. 3 and 4 Knockertotan (Portrush Rural, Antrim) this changes in 1911 to No 1 Benjamin Connor also in 1901 is another Conner family in No 6 Urbalreagh (Portrush Rural, Antrim) above Knockertotan closer to north coast
see the townland locations
https://www.townlands.ie/antrim/dunluce-lower/ballywillin/knockertotan/from
https://www.townlands.ie/antrim/dunluce-lower/ballywillin/portrush/same source for Urbalreagh (Portrush Rural, Antrim
in 1911 in 1911 John Conner has gone his widow Esther remarries as Easter Maryhas children but keeps her own Conner children and the family lives at 3 in Ballycraig, Lower (Portrush Rural, Antrim) BUT THEIR NAME NOW is CONNOR
there are more Connor to exclude in these two census.
from all these websites you can see how close there are for churches probably Portrush, Portballintrae, Bushmills to the north Magherabuoy Presbterian to the west, Coleraine too far
Finally in that spelling the earliest Francis Conner found is son of Patrick b 1679 see
https://www.ancestryireland.com/ Because of the brevity and minimal presentation of your ‘Another Connor Search’ I replied to best of my ability in the time available to me, because of your minimal approach I became irritated in my work. I was expecting you to give me a detailed reaction to each of the four individuals as to how you had selected them and had some further information to offer and you had already started to looking up Coleraine and Ballinderry sources for their deaths or marriages etc them yourself . or had already done so.. So I offer my apology for my unrealistic expectations, so now perhaps you will consider replying to present and future other readers as to whether if Francis Conner is a good or bad clue for you. And others might join in your quest but i retire now from the query.