When Gödel published his paper on the incompleteness theorems, nobody had identified "Peano arithmetic" as a particular theory. Gödel instead referred to a system based on Principia Mathematica. It quickly became clear to Gödel and others that the same principles would apply to any $\omega$-consistent, effective, consistent theory that includes a small amount of arithmetic.
But nobody had studied Peano Arithmetic in particular before the incompleteness theorems were published. The history of the development of logic is an epitome of how many mathematical areas are discovered: not in an orderly way, but in fits and starts with researchers working independently, duplicating some results and taking a long time to fill in what are later considered basic topics. In particular, the underlying topic of "first order logic" was not really clear until the late 1940s or 1950s, when people had finally had enough time to digest the various results of the previous 50 years and organize them into a coherent system.
Before Gödel, Peano and Dedekind had both proposed axiomatic systems for arithmetic. These systems were second-order, however - they presupposed a notion of "set", which they did not axiomatize. So proving things from these axioms requires using not only the axioms, but also principles about set existence. We often call these the Peano axioms or the Peano postulates to differentiate them from Peano Arithmetic.
One of the key differences between these systems and the theory called "Peano Arithmetic" is that Peano Arithmetic is fully axiomatized, without assuming any background theory of sets. So it is possible to write formal proofs of many arithmetical statements fully from the axioms of PA, unlike the Peano axioms.
I am not sure of the first appearance of the specific axioms for Peano Arithmetic in print, or who coined that name for it. If I had them at hand, I would look at Kleene's Introduction to Metamathematics and Robinson's paper establishing Robinson Arithmetic, both from the 1950s, to see if they give any clue.