Just as puppies are not a breed of dog, but (as anyone who ever had one quickly found out) baby dogs, new potatoes are not a type of potato, but immature potatoes of any variety. Contrary to popular belief, new potatoes are not necessarily red-skinned potatoes, any more than puppies are necessarily beagles. It may be that red-skinned potatoes are the most popular variety harvested early and sold as new potatoes, but you can certainly also purchase fully mature red-skinned potatoes.
New potatoes - or early potatoes in Britain - have not had as much time to convert their sugar into starch, which is probably what qualifies them for your particular diet. Their skin is not very well developed, and will be very thin in places where it has not rubbed off entirely. Their high moisture content makes them very waxy and renders them poor candidates for either baking or frying. The best cooking options are boiling or steaming.
The reason you may be having trouble finding them in the dead of winter is that new potatoes are shipped from field to market directly after being dug, and they are most readily available in spring and early summer. Mature potatoes have a longer harvest season and may spend many months in storage before coming to market, which is why you can find some types of potatoes year-round in your grocery store. But you may have to wait a few months before you can find new potatoes.
Related Article: A Potato Primer