Merry always had 1 sword: the sword from the Barrow-Downs. Galadriel gave him a silver belt.
Here is what happens in the book. (Emphasis mine)
For each of the hobbits he chose a dagger, long, leaf-shaped, and keen, of marvellous workmanship, damasked with serpent-forms in red and gold. They gleamed as he drew them from their black sheaths, wrought of some strange metal, light and strong, and set with many fiery stones. Whether by some virtue in these sheaths or because of the spell that lay on the mound, the blades seemed untouched by time, unrusted, sharp, glittering in the sun.
'Old knives are long enough as swords for hobbit-people,' he said. 'Sharp blades are good to have, if Shire-folk go walking, east, south, or far away into dark and danger.' Then he told them that these blades were forged many long years ago by Men of Westernesse: they were foes of the Dark Lord, but they were overcome by the evil king of Carn Dûm in the Land of Angmar.
The Lord of the Rings - Book I, The Fellowship of the Ring: Chapter VIII: Fog on the Barrow-downs
Then Merry looked for his sword that he had let fall; for even as he struck his blow his arm was numbed, and now he could only use his left hand. And behold! there lay his weapon, but the blade was smoking like a dry branch that has been thrust in a fire; and as he watched it, it writhed and withered and was consumed.
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dúnedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.
The Lord of the Rings - Book V, The Return of the King: Chapter VI: The Battle of the Pelennor Fields
The sword of the Barrow-downs was used to injure the Witch-king.
Merry's gift from Galadriel:
The Lady bowed her head, and she turned to Boromir, and to him she gave a belt of gold; and to Merry and Pippin she gave small silver belts, each with a clasp wrought like a golden flower.
The Lord of the Rings - Book II, The Return of the King: Chapter VIII: Farewell to Lórien
In the film it is different, and thus your points are correct. Merry (and Pippin) both received swords from Galadriel, although, the sword he used to stab the Witch-king is the sword from the Barrow-downs, not Galadriel's gift.