Comment by AdAbsurdum

I think the relevant quotes are these:

"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.