Mary Barnes is believed to have been hanged 25 Jan 1662/3 -- but there is no firm documentation that I know of. We know that she was condemned to die, and we know that the Greensmiths were condemned at the same time, but details on the actual execution are hard to establish with great confidence.
You're quite right about the difference in record-keeping between Hartford and Salem. I'm related to many of the central figures in the Salem witch trials, including my 7th-g-grandfather, Deacon William Fisk, who was one of the 12 jurors who signed the Letter of Regret. There's a plethora of great documentation to sift through on Salem. Conversely, the Barnes case is hard to piece together because there's scant reliable info out there and a whole lot of erroneous info and unfounded speculation that keeps getting repeated.