Thanks, Elwyn and aghadowey.
I think it's pretty certain that the church called the Old Presbyterian Meetinghouse in 1868 is the disused NSP/Unitarian church. The marriage of Bristow Charters and Anne Murphy there (19 June 1868) was conducted by Rev. James McFerran. In the Belfast and Province of Ulster Directory for 1870 he's listed as the Unitarian minister in Antrim.
I'm inclined to discount 2nd Antrim (and indeed 1st Antrim). No doubt they were described as meeting houses, but this church looks like the best candidate to be The Meeting House, so that its green might be expected to be recognized, in a BNL death notice, under the label Meeting-House Green. And it certainly had/has a graveyard!
For what it's worth: Mary Mackey, who spearheaded the conversion to Methodism of the Mackey siblings, Mary, Martha, and Alexander, was previously a Unitarian. (See reply #2 on this topic.) Martha became the mother of Rev. Thomas McLorinan, one of the people said to have been buried in the McLorinan family burial ground. It's surely likely that this was the family church of the Mackeys before the conversion.
I wasn't hoping for church burial records. I was hoping someone might find McLorinan graves in the graveyard. And of course Mackey graves would be interesting too.