First, there is no "history" to speak of, as the tale of Arthur and Excalibur is fictional.
Otherwise, after looking at the Excalibur article on wikipedia, it appears that:
- The Sword in the Stone is the sword that proves Arthur's right to rule after Uther Pendragon;
- Caliburn is always an alternate name for Excalibur;
- According to the Vulgate Cycle (A 13th-century set of poems in french), Excalibur is explicitly identified as the Sword in the Stone;
- According to the Post-Vulgate Cycle (A revision of the Vulgate Cycle, with a lot of changes), Excalibur is the sword given by the Lady of the Lake, and was returned to her after Arthur's passing;
- Thomas Malory later compiled a number of Arthurian legends in his 1485 compilation called Le Morte D'Arthur. In it, he mentions both versions of Excalibur's origin. Thomas Malory's work has been the main source for many modern Arthurian authors;
In the end, Excalibur is the Sword in the Stone in some versions of the story, and in other versions it is not. In Fate/Stay Night, the latter option was chosen.
However, the act of using the name "Caliburn" for a sword distinct from "Excalibur" seems uncommon. I suspect however that it is done so that it can be referred to by name instead of always having to call it "The Sword in the Stone".