I'm trying to find a record that states clearly one way or the other. If it was Palmyra then that creates obvious problems with the JSH canonized account of the first vision.
Apparently the official Smith family history says Palmyra, as well as Lucy's patriarchal blessing, but I haven't been able to find and confirm a source for either.
familysearch.org has Lucy born in Manchester. Har Har Har @ TSCC if there is no source for that.
The Smith home was right on the town boundries. Smith place is closer to Palmyra Village than Manchester Village but is actually in Manchester township. Most people consider it Palmyra. How's that for adding to the confusion?
Smith family also lived on Main Steet in Palmyra when they first arrived. They owned a cake shop. In 1822 they moved to a cabin in Palmyra a few feet from Manchester and later built a house over the Manchestre line.
This URL might help. Boundry lines changed in 1823. Perhaps checking Ontario County records would help. Part of the Town of Farmington where the Smiths lived also became part of Manchester.
I think it's pretty obvious from critics and apologetics alike that they were on the Palmyra side of the line until at least 1822, when the cabin and/or frame house was completed.
The question is whether or not Joseph Smith thought they were in Manchester at the time.
From what I've found, the Smiths didn't buy the Manchester property until August 1820 since the guy who sold it to them didn't receive the power of attorney until July 14, 1820. Then their names appear in the Manchester census sometime between August 1820 and February 1821.
Apologists like to claim that the Smiths might have thought they were living in Manchester while in the house where Lucy was born, which is about 50 feet on the Palmyra side of the line. Of course this isn't supported by their names showing up all over the Palmyra records until 1822. The more I can find that confirms they knew they were in Palmyra, the better.
The benefit of finding out if Lucy was born in Palmyra is that it sets a definite earliest possible move date at her birthday.
For those who are fuzzy on the subject, The Joseph Smith History in the Pearl of Great Price says that the religious excitement and First Vision (in the Spring of 1820) occurred in the second year after their move to Manchester. This can't be true if they didn't move to Manchester until 1822 or 1823. Ironically the only religious excitement around that time was 1824-1825. Coincidence?
The town and county lines are pretty iffy around the Smith farm even now. Part of what the Church designated as the Sacred Grove is in Palmyra and part in Manchester.