Hi Leslie,
I added the source for Barnabas Harlow's birth in 1784 to his profile and connected him to his parents. It is behind a paywall on American Ancestors, so I transcribed the text for you into the bio.
For Elizabeth, I think you have her parents names right. The records for Liverpool on American Ancestors show Samuel Kinney married Sarah Howard in Liverpool on April 25, 1776, He was the son of [[Kinney-1746|Elisha Kinney]] and Jane or Jean his wife. She was the daughter of Francis and Elizabeth Howard. Samuel and Sarah are mentioned several times in Simeon Perkins Diary, and other sources for Liverpool and Milton.
The records for Samuel and Sarah (Howard) Kinney don't show any daughters, just 4 sons all born 1778-1786, but these records do have some gaps, and I think you're reading Elizabeth's death record right. The handwriting for her mother's surname is pretty bad, but Samuel is clearly a K. There were various spellings for the Kinney/Kenney/Kinnie families among the New England Planters. Most of them settled in Barrington and Yarmouth, only the one family went to Liverpool. I'm quite certain there were no others with similar names who could provide a viable alternative for Elizabeth's birth in Milton at that time.