Yes, those are called Unipak and I believe this Dionne Warwick album may have been the first issued in one. A company called Sherwood Litho introduced the design in early 1968. They were pretty rare to find after 1973. I personally never liked them, they tend to have seams come apart and were clumsy to remove the record from.
I think I know what you mean about the cover description. I'm looking at a cover that fits your description. It's "The Best Of Tommy James & The Shondells" (which you uploaded the front cover of here). The front cover is also single ply card and the record opening is on the left side after you open the front cover. Right on the crease where the front cover folds is the following text: "UNIPAK U.S. Patent No. 3,426,960". So maybe it's called a unipak cover.
Yes, the cover image I uploaded is gold, metallic but not foil. Also, I don't recall the term, it's like a gatefold cover but the front is a single ply card. The front and back are the same, the image of your back cover is inside the front cover.
Added red and black label scans
Not sure what they mean by "Stereo-Monic" though
Added front cover variant, back cover and label images. The first issue cover was a shiny gold (gold foil?) color, which I think front cover {Image #599090} is. The first issue label was the red and black label.