In the broadcast world, you see somewhat different distinctions:
Producers set overall objectives for show content, but generally don't determine exactly how those objectives get met. They operate at the most general level, making sure the content elements are lined up.
Directors call the show, coordinating camera, audio, video switching, and stage functions. They have a "big picture" logistical function, making the producer's objectives do-able on a creative level.
Technical Directors actually operate the production switcher, and are mainly focused on the "right now" of switching.
Operators are minimally technical folks who know how to run a particular class of equipment: camera, videotape, Chyron, slomo, etc.
Engineers are the most technical, and are responsible for design, installation, and maintenance of the systems. They are also the people who make the objectives of the producers, and the methods of the directors, physically practical.
There's almost always tension amonst these groups: producers get frustrated with engineers for crimping their vision; engineers get frustrated with producers for their practical cluelessness.
You also don't want to stand in the way between an engineer and free food.
-- Jeff