How did this area come to be called Kamakura?

There are various theories about the origin of the place name, Kamakura. The following is the most probable.

　The central part of Kamakura is surrounded by hills on three sides--north, east, and west--and the fourth, south side is open to the sea. This lay of the land was likened to a stone or earthenware cooking stove used in the old days, kamado (竃) in Japanese, and also to a warehouse, kura (倉), because both kamado and kura have one open side and three closed ones. The combined name, kamado kura, it is said, gradually turned into the name, Kamakura.

　Another theory is based on a legend that dates back to the seventh century.
　Fujiwara no Kamatari (藤原鎌足, 614-669), the founder of the Fujiwara family that prospered in the Heian period (794-1185/92), stopped by at Yuigahama (由比ガ浜) on his way to Kashima Jingu Shrine (鹿島神宮) in present-day Ibaraki Prefecture, where he was to pray for peace and order in the eastern part of Japan. That night he had a dream in which an old man appeared and promised to support him. The next morning he found at his bedside a spear with a sickle-like blade, the type known as kamayari (鎌槍). Kamatari respectfully enshrined it at a spot in Okura (大蔵). The kama in kamayari and the kura in Okura are said to have produced the name, Kamakura.

*Kamatari Inari Shrine, which is closely connected with Fujiwara no Kamatari, within the grounds of Jomyoji Temple in Jomyoji. A plaque next to the Shrine reads as follows. 　As an infant, Lord Fujiwara no Kamatari was given a sickle by the Inari no Kami.　By keeping the sickle with him at all times as a charm, Lord Fujiwara no Kamatari was granted the protection of the deity. 　In 645 A.D. he joined Nakano-oe-no-oji (later the Emperor Tenchi) to vanquish the noble Soga-no-Iruka and thus succeeded in fulfilling his greatest wish. 　Lord Fujiwara no Kamatari then headed for the eastern provinces of Japan in 646, stopping at Yui-no-sato in Sagami. That night, he received a message from the Inari no kami, who told him, "I gave the sickle to you and have protected you ever since, but now you have achieved your greatest wish by defeating Iruka, so you must return the sickle to me and to the land." Doing as he was bidden, Lord Kamatari buried the sickle, and this action and his prayers offered up to the god were the beginning of the Kamatari Inari Jinja. 　The place name Kamakura is said to signify the location where Lord Kamatari buried the sickle, or "kama." 掲載許諾済

When and in what documents did the place name Kamakura appear in early history?