In J.R.R. Tolkien’s The Lord of the Rings, the wizard who returns as Gandalf the White is Gandalf.
Gandalf the Grey confronts the Balrog on the Bridge of Khazad-dûm while the Fellowship escapes Moria. He falls into the abyss with the creature and later explains that they continue fighting until they reach the peak of Zirakzigil. After defeating the Balrog, Gandalf dies and is sent back to Middle-earth with greater authority.
His return as Gandalf the White changes his position among the Istari. He takes Saruman’s place as the chief opponent of Sauron’s forces after Saruman turns traitor. Gandalf’s white clothing also visually contrasts with the grey robes associated with his earlier identity.
Gandalf is not a new person after the transformation. He remains the same Maia named Olórin in Valinor, but his mission and power are renewed. Saruman, Radagast, and the two Blue Wizards are separate members of the order.