You just want lightweight running shorts. Paper thin nylon. Elastic waist. Some have pockets, some don't -- maybe an inside pocket for a car key or credit card. They weigh nothing.
Rather than point you to some $$$ option, all of the running companies (Nike, Adidas, Brooks, New Balance, and on and on) make them in various lengths (3 inch, 5 inch, 7 inch, etc). This is the kind of thing you can easily pick up at the discount outlets (TJMAXX, Marshalls, etc.) -- despite the current trend in men's shorts for mid-calf capri length. From what you describe, the 5} version should be fine -- about the length of boxer shorts.
Most will have liners, which you could cut out if you like, although it's probably not necessary.
Brooks Essential Run Short