Arthurian legend is essentially combined from a wide variety of sources, and there doesn't seem to be a definitive answer.
The first narrative account is from Geoffrey of Monmouth in Historia Regum Britanniae (History of the Kings of Britain), 12th Century. He wrote, in Latin, of a sword called Caliburnus, which was made on the isle of Avalon.
Wace wrote Roman de Brut (12th Century), which is described as "an Old French translation and versification of [Monmouth]". According to Wiki, the sword is called Calabrum, Callibourc, Chalabrun, and Calabrun (with alternate spellings such as Chalabrum, Calibore, Callibor, Caliborne, Calliborc, and Escaliborc).
Chretien de Troyes wrote, in Perceval, (again in the 12th Century), of a sword called Escalibor or Excalibor.
Robert de Boron (late 12th/early 13th Century), in Merlin, wrote of the Sword in the Stone, and about the idea of only "one true king" being able to retrieve the sword. It isn't confirmed that this sword is Excalibur, but later versions took this story and called the sword Excalibur.
Later, in the Post-Vulgate Cycle (13th Century), French writers attempted to bring together some of the scenes and characters of the legends (Wiki calls it a "rehandling of the earlier Vulgate Cycle"). In this version, the sword is called Excalibur, and is given to Arthur by the Lady of the Lake.
In Le Morte d'Arthur, by Thomas Malory, he gives both versions, and calls both Excalibur. Arthur receives the sword three different times in Le Morte, in three different ways (the stone, the lake and once in a battle).
Finally, according to the 15th Century Alliterative Morte Arthur, there is a sword called Clarent, a fragile sword designed for ceremonial purposes rather than fighting. The internet (though I haven't found an original source for this yet) associates this with being the Sword in the Stone.
In conclusion, there isn't a clear answer. The stories evolved quite a bit, and the sword that some writers took to be Excalibur was taken by others to mean Clarent.
However, it does seem that:
- Excalibur and Caliburn do seem to be the same sword. It is a linguistic evolution. Both are taken to be Arthur's "main" sword, regardless of writer or name.
- The Sword in the Stone, and the one from the Lake, may or may not be the same sword.
- Clarent is not Excalibur, and is a different sword, wielded by King Arthur at points (and later, Mordred). It may be the Sword in the Stone, but is not the sword from the Lady in the Lake.