Another British English answer, with a few more terms:
- Body abseil: rarely used term, any abseil not using a abseil/belay device or carabiner
- Classic abseil: Rope goes between legs, round the thigh, across the chest, over the shoulder, across the upper back, down the arm. This is also known as Dülfersitz, but that's not a common term.
- South African abseil: Needs two ropes, one goes under each armpit, cross behind the back, come over the hips and between the legs, and then both together up to the hand.
- Body Belay: A similar technique for belaying with the rope wrapped around you instead of a belay device. Still a popular technique for easier ground in winter, often combined with a bucket seat. The rope comes along your arm, behind the small of your back, then wraps around your other arm.
Body abseils are taught as emergency techniques. When I did my Mountain Leader ticket (10+ years ago) you were expected to demonstrate one or both techniques on c. 60 degree ground, but I've used both on free hanging abseils too, to see what it was like. Gloves are strongly recommended especially for skinny modern ropes, and even if done correctly it's going to hurt a bit on steep or overhanging ground. You are usually taught to use the South African where possible, as it is safer and less painful. The classical is reserved for when you need to go a whole rope length in one go. These are very much the dominant techniques, most instructors would call other ways of wrapping the rope "wrong".
I can't tell exactly where the rope goes in that photo. It looks like it goes up the arm, across the back, down the other arm, but if so, then I don't think it is a safe technique. That is kind-of the route used for a body belay, but that only works properly with the rope going down (as for belaying a second) not up (as for an abseil). I think it is likely that there is more rope, hidden in the folds of his jacket. In particular the flash of white near his navel might be more rope - but I can't figure out how it's routed.