These practitioners who claim direct or indirect descent from the works of Gerald B. Gardner call themselves Wiccan. On the West Coast alone, in addition to Gardnerians, there are British traditions called Kingstone, Silver Crescent, Majestic, Georgian, Central Valley Wicca. Some say we all owe a debt to the British, specifically to Gerald B. Gardner, Doreen Valiente, and others. We can trace some of practices, such as Elemental correspondences, to Western ceremonial magic. The Elizabethan John Dee spoke of Watchtowers.
Pentagrams appear in the Key of Solomon.
This heritage appears chiefly within the broader Gardnerian revival. I certainly agree that all Craft traditions in the U.S. today owe a debt to Gardner, et al. The influence of these British traditions is evident in Reclaiming practice. We purify ourselves and our working space. We work in a sacred circle of our construction.