[cubefox]

Pure terminology. In some areas "Peano Arithmetic" is short for "first-order Peano arithmetic". In first-order logic, Peano's induction axiom can't be properly formalized, which means the resulting theory isn't categorical. For other people "Peano arithmetic" just describes the usual Peano axioms in natural language, which includes the proper induction axiom. Which can only be formalized by using at least second-order logic. That theory is categorical.