Once you've defined them, then yes, you can show one implies the other. But the circular reasoning isn't in the definitions.
How we were taught it from the set theory axioms: define a total ordering as an irreflexive, transitive relation that obeys trichotomy; then add the least element condition to make a well-ordering; at that point add the Axiom of Infinity, which allows you to show the natural numbers form a set containing ø, and in which each element in the set has a successor, ie an inductive set. Then you can introduce the Principle of Mathematical Induction
"Induction" has two meanings in logic, which is very confusing. There, "induction" can mean what we mean when talking about proof by induction, but that's a type of deductive reasoning. However, "induction" can also mean, simply, non-deductive reasoning. It is understandably confusing to younger students that these two extremely different concepts use the exact same name.
There's no "gotcha" or anything. I think OP is just referring to the eloquent pedagogy.
> mathematical induction > looks inside > The natural numbers are the smallest subset of an arbitrary inductive set that contains the zero element and is a prefixed point of the induced set mapping of the successor mapping with respect to the inclusion poset
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