On December 7, 1873, he wrote to Dedekind that he thought he’d finally succeeded: “But if I should be deceiving myself, I should certainly find no more indulgent judge than you.” He laid out his proof. But it was unwieldy, convoluted. Dedekind replied with a way to simplify Cantor’s proof, building a clearer argument without losing any rigor or accuracy. Meanwhile Cantor, before he’d received Dedekind’s letter, sent him a similar idea for how to streamline the proof, though he hadn’t worked out the details the way Dedekind had.
"Dedekind quickly replied that...he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted.
[...]
Weierstrass had been most excited about the proof that algebraic numbers are countable. (He would later use that result to prove a theorem of his own.) So Cantor chose a misleading title [for his paper] that only mentioned algebraic numbers.
[...]
Writing his paper, Cantor put the proof about algebraic numbers first. Below it, he added his own proof that the real numbers cannot be counted — Dedekind’s simplified version of it, that is."
So the first proof -- the one the article was titled after -- was completely created by Dedekind.