If natural numbers are defined as $\mathbb{N} = \{1\} \cup \{n + 1 \mid n \in \mathbb{N}\}$, and we know that $P(1)$ and $\forall n \in \mathbb N,P(n) \implies P(n+1)$, then $S = \{ n \mid P(n)\} = \{1\} \cup \{n + 1 \mid n \in S\} = \mathbb N$, meaning that $\forall n \in \mathbb{N}, P(n)$.
In this case why do we need induction (or the equivalent well-ordering principle) as an axiom?
Your definition is circular, you use $\mathbb{N}$ to define $\mathbb{N}$. To construct the set $\mathbb{N}$, use the Peano's Axioms, and the fifth axiom is percisely the Induction Principle: If $S \subseteq \mathbb{N}$ is a set such that $0 \in S$ and $n \in S \Rightarrow n+1 \in S$ then $S=\mathbb{N}$.
The Induction Principle and the Principle of well-ordering are equivalent.