A design engineer was an individual who created the blueprints and schematics for a starship. These engineers were assigned to a team, which belonged to an engineering group.
According to the USS Enterprise-D dedication plaque, Captain Andrew Probert was the design engineer for this starship. (Star Trek: The Next Generation, set artwork)
In 2363, design engineers assigned to the Theoretical Propulsion Group were responsible for the creation of warp core schematics for the Galaxy-class starships. One of these engineers, Doctor Leah Brahms, was assigned to team seven as a junior member. In four years, she was a Senior Design Engineer for this group. (TNG: "Booby Trap", "Galaxy's Child")