In the script of "Playing God" the Nash was simply identified as a personnel transport.  In the script of "Accession" the Nash specified as a Bajoran transport.  It was called a Federation transport in the script of "Trials and Tribble-ations"  and a Starfleet transport in "Chrysalis".  The Nash was named by the studio modelers for Erik Nash, a motion control technician at the special effects company Image G.
A legible picture of the studio model labeled as the Nash was first seen in the Cinefantastique Magazine December 1994 issue. The name and registry are derived from there. The Nash studio model was only seen docking with DS9 "upside-down", at least in respect to the models previous appearance as the USS Jenolan in TNG: "Relics". The name and registry decals of the Nash however indicate that this was the correct orientation for this ship.