The one you describe first ("Gable roof, a door and 2 window in the middle section") is "the" Moulton barn that is commonly called such, was featured in the movie "Spencer's Mountain", and is much photographed. It was built by Thomas Alma Moulton. T. A. Moulton's brother John also built a barn on Mormon Row, and it is the second one you describe. So they are both Moulton barns, but if there is a "real" one, it is T.A.'s.
I live in the next valley to the west of Jackson Hole, and know some of the Moulton's. Candy Moulton wrote a book (Legacy of the Tetons: Homesteading in Jackson Hole) that describes the barns' history, and I used this as a source.