I started reading James Knowles' The Legends Of King Arthur And His Knights. The story of the sword in the stone and Excalibur is related in chapter II. Apparently, they were two different swords. Here are the relevant excerpts:page 41 in the paper wrote:This sword and the Lia Fáil later ended up in Europe, being “the sword in the stone,” Excalibur.
Sword in the stone
And as they prayed, there was seen in the churchyard, set straight before the doorways of the
church, a huge square stone having a naked sword stuck in the midst of it. And on the sword was
written in letters of gold, “Whoso pulleth out the sword from this stone is born the rightful King of
Britain.” At this all the people wondered greatly; and, when Mass was over, the nobles, knights, and princes ran out eagerly from the church to see the stone and sword; and a law was forthwith made that
whoso should pull out the sword should be acknowledged straightway King of Britain.
Then many knights and barons pulled at the sword with all their might, and some of them tried
many times, but none could stir or move it.
[...]
So he [Arthur] rode and came to the churchyard, and alighting from his horse he tied him to the gate, and
went to the pavilion, which was pitched near the stone, wherein abode the ten knights who watched
and kept it; but he found no knights there, for all were gone to see the jousting. Then he took the sword by its handle, and lightly and fiercely he pulled it out of the stone, and took his horse and rode until he came to Sir Key and delivered him the sword.
Excalibur
Then Arthur said to Merlin, “I have need now of a sword that shall chastise these rebels terribly.”
“Come then with me,” said Merlin, “for hard by there is a sword that I can gain for thee.”
So they rode out that night till they came to a fair and broad lake, and in the midst of it King Arthur
saw an arm thrust up, clothed in white samite, and holding a great sword in the hand.
“Lo! yonder is the sword I spoke of,” said Merlin.
Then saw they a damsel floating on the lake in the Moonlight. “What damsel is that?” said the king.
“The lady of the lake,” said Merlin; “for upon this
lake there is a rock, and on the rock a noble
palace, where she abideth, and she will come
towards thee presently, thou shalt ask her
courteously for the sword.”
Therewith the damsel came to King Arthur, and
saluted him, and he saluted her, and said, “Lady,
what sword is that the arm holdeth above the
water? I would that it were mine, for I have no
sword.”
“Sir King,” said the lady of the lake, “that sword is mine, and if thou wilt give me in return a gift
whenever I shall ask it of thee, thou shalt have it.”
“By my faith,” said he, “I will give thee any gift that thou shalt ask.”
“Well,” said the damsel, “go into yonder barge, and row thyself unto the sword, and take it and the
scabbard with thee, and I will ask my gift of thee when I see my time.”
So King Arthur and Merlin alighted, and tied their horses to two trees, and went into the barge; and
when they came to the sword that the hand held, King Arthur took it by the handle and bore it with
him, and the arm and hand went down under the water; and so they came back to land, and rode
again to Caerleon.
[...] but Arthur, turning to his knights, fought ever in the foremost press until his horse was slain beneath him. At that, King Lot rode furiously at him, and smote him down; but rising straightway, and being set again on horseback, he drew his sword Excalibur that he had gained by Merlin from the lady of the lake, which, shining brightly as the light of thirty torches, dazzled the eyes of his enemies.
