The story of the sword in the stone survives in the “Story of Merlin” ascribed to Robert de Boron. It is dated to the end of the 11th century or to the beginning of the 12th century. All later accounts, including Malory’s, are almost certainly taken directly from Robert de Boron’s account. But de Boron does not gave any name to this sword.
Artthur’s sword was already known in romances as Caliburn or Escalibor. (Malory names it Excalibur.) So many who read or heard Robert de Boron’s story would be likely to assume that the sword in the stone was identical to Caliburn/Escalibor. Indeed, that may even have been Robert de Boron’s intention.
The chronicle accounts of Arthur, imply that Arthur was brought up in his father’s court and was made king normally when his father died. This is implicitly indicated in some romances: in the “Parzival” by Wolfram von Eschenbach and in “Diu Krône” by Heinrich von dem Türlin. In these versions the origin of Caliburn/Escalibor is never stated, other than the Geoffrey of Monmouth and those who adapt his chronicle state that it was forged in Avalon.
The sword in the stone is explicitly identified with Calibur/Escalibur in a later sequel to Robert de Boron’s story, often known today as the “Vulgate Merlin” (because this sequel is part of a group of romances know as the “Arthurian Vulgate Cycle”). A translation into late Middle English is found on the web where any can read this identification. See http://www.lib.rochester.edu/camelot/teams/barevfr.htm , beginning with line 132.
Sir Thomas Malory uses this material in his “Le Morte d’Arthur”. See http://www.sacred-texts.com/neu/mart/mart008.htm :
“Sir, said Merlin to Arthur, fight not with the sword that ye had by miracle, till that ye see ye go unto the worse, then draw it out and do your best.”
“Then he drew his sword Excalibur, but it was so bright in his enemies’ eyes, that it gave light like thirty torches.”
About the same time as this sequel was being written, another alternate sequel was written by another author. This alternate account of Arthur's early reign is often known today as the “Post-Vulgate Merlin” (as it was written after most of the “Arthurian Post-Vulgate Cycle”). In this account, unlike the account in the “Vulgate Merlin”, Arthur gets his sword Caliburn/Escalibor from a lake fay. The “Post-Vulgate Merlin” was adapted in medieval Spanish as the “ Baladro Merlin'', and this has in turn been translated in modern English on the web. See http://members.terracom.net/~dorothea/baladro/index.html . The actual account of Arthur getting his sword has dropped out of the text. It should have occurred immediately following chapter 21. The sword from the lake is identified with Escalibor towards the end of chapter 22 where Arthur asks its name and the fay names it as Escalibor, rendered by the translator as “Excalibur”. Malory, in his “Le Morte d’Arthur” also gives this alternate version. See http://www.sacred-texts.com/neu/mart/mart024.htm and http://www.sacred-texts.com/neu/mart/mart029.htm .
In neither the “Post-Vulgate Merlin” or in Malory is it definitely stated that the sword broken by Arthur in battle with King Pellinor is the sword from the stone. One may, of course, assume that it is. And neither the ”Post-Vulgate Merlin” or any accounts derived from it name this sword that was broken in battle.
The answerer who claimed that the sword in the stone was not Caliburn/Escalibor, and that one should read the old romances, seems not to have read them himself. Two alternate versions of the origin of Caliburn/Escalibor/Excalibur have come down to us and Malory includes both of them in his work despite the fact that they contradict one another, unless one assumes that both swords were named Calibur/Escalibor/Excalibur, which does seem unlikely.
(But, indeed, both these tales seems somewhat unlikely. They are fantasy stories.)
Another romance, known as the English “Alliterative Morte Arthure” mentions, besides Arthur's sword Caliburn, another sword owned by Arthur which was named Clarent and was a ceremonial sword inherited from his ancestors which Arthur had never carried into battle. See http://www.lib.rochester.edu/camelot/teams/allitfrm.htm , line 4192 and following. I have read some ignorant commentators who claim that this sword, which Arthur never carried into battle, should be identified with the sword in the stone, which broke in Arthur’s battle with King Pellinor. This is just another false claim.
In short, the “Vulgate Merlin” reasonably identifies Arthur's famous sword Caliburn/Escalibor with the sword in the stone and so Caliburn/Escalibor is Arthur’s first sword. But the “Post-Vulgate Merlin” instead claims Arthur received Caliburn/Escalibor later from a lake fay after breaking another sword, which might be the sword from the stone (or might not). This lake fay is not to be identified with the main Lady of the Lake who imprisoned Merlin and who brought up Lancelot. She dies soon after at the beginning of the story of Balaain/Balin. But Tennyson and some later modern writers have confused them and also often split the proper Lady of the Lake into two persons.
Clarent presumably has nothing to do with this. The author of the “Alliteratve Morte Arthure” may not have known or may not have accepted Robert de Boron’s account of the sword in the stone. If he did accept the tale, he probably would have accepted this sword as Caliburn/Escalibor. He certainly does not mention that Clarent was taken from a stone, which he would probably have done if he had thought so, since he wishes to emphasize how much Arthur prizes Clarent.
Also neither Robert de Boron nor any later account taken from his account tells anything about the origin of the sword in the stone. That the sword in the stone was a sword of Arthur’s father is the invention of some modern novelists. This idea also appears in John Boorman’s film “Excalibur” where Boorman merges the two accounts by having the sword broken in battle with King Pellinor rejoined by the magic of a lake fay. But this is modern invention, not found in any medieval account. (Though there is an account of Perceval having a sword that will only break in one peril, and one account states it was magically reforged after being plunged into a lake.)
Also in no medieval account is it said that the sword was placed in the stone by Merlin. The miraculous appearance of the stone in the churchyard is ascribed to a miracle of God.
See http://www.maryjones.us/ctexts/kaledvwlch.html for a late Welsh version mainly taken from Robert de Boron’s story in which the sword in the stone is identified as Kaletvwlch, which is the name of Arthur's sword in Welsh tales, corresponding to Calburn/Escalibor in French and Latin texts. It is used for these French forms in Welsh translations of Latin and French texts.
In many romances Arthur's nephew Gawain carries Arthur's sword Caliburn/Escalibor as his own sword. The “Vulgate Merlin” explains that Arthur bestowed Caliburn/Escalibor on Gawain when he was knighted. Arthur keeps as his own sword, a sword named Marmiadoise which he had won in battle from King Rion (whom Malory names as Rience. In the “Prose Lancelot” it is mentioned that Arthur has a sword of his own, other than Caliburn/Escalibor which Gawain is portrayed as bearing, This other sword is called Sequence, but we are told nothing of its origin.