Peep sights offer two advantages.
When used, properly, one advantage is reducing the the number of sights to line up from two to one. Being able to focus solely on the front sight is a huge advantage, all the more so since it's the rear sight that so many of us with old eyes find to be the fuzziest. The barrel mount peep will offer this advantage, but you have to make a conscious effort to look though the sight, same as any other peep, and avoid the temptation to line up the front sight in the center of the peep. This may take a little practice, but it's doable. Have barrel mounted peeps on a couple of rifles.
The other advantage of the peep is the sharpening effect on the front sight. However, this varies with the combination of aperture size and distance of the peep from your eye. The most effective peeps in this regard are peeps closest to your eye, because they allow the use of very small apertures. This enhances the sharpening effect. The farther away from your eye, the less effective since you must use a larger aperture to get a sufficient field of view. In this regard, the barrel mounted peep offers minimal sharpening, if any. You'll need to use a fairly large aperture, usually around .125" as a minimum to get a reasonable field of view and that's not going to do much to sharpen things up.
For me, the big advantage pf the Skinner barrel mount peep is that it allows me to outfit a peep on a rifle that is not set up to take a receiver mounted peep. As I said, I have them on some older rifles. Another plus for the barrel mount peep is that you can usually use the factory front sight. When adding receiver peeps to the 10/22, you usually have to go to a taller front sight and, frankly, getting that factory front sight out on some individual 10/22s can be a real pain.
On the 10/22 with its short barrel, there's also the issue of skimpy sight radius. That's another factor in iron sight effectiveness. A receiver mounted peep would increase sight radius on the 10/22 as well as add some sharpening and thus do a better job of squeezing out accuracy from the 10/22. Unless you plan to take down your 10/22 on a regular basis, I'd opt for the receiver peep, even if it meant some re-sighting in, now and then. Your call, though. Either will work.