I think you are missing the point of the Peano Axioms.
It postulates that there is a set $\,\mathbb{N}\,$ which
is by convention called the set of natural numbers. We
are given that zero is a natural number and is by convention
denoted by $\,0.\,$ In order to avoid confusion and emphasize
its nature, perhaps it would be better to use a distinct
notation such as "$0$". The successor of zero is a natural
number by convention denoted by "$1$". This conventional
defining property of "$1$" as the successor of "$0$" in
conjunction with other definitions and the Peano Axioms
leads to all of its properties. Similarly for all of the
other natural numbers. Each natural number is defined to
be the successor the the previous number. In other words,
Only the number zero is initially given and all of the rest
of the natural numbers are determined as the repeated
successors of zero. For example,
$$ \text{"}1\text{"} :=S(\text{"}0\text{"}),\;
\text{"}2\text{"}:=S(S(\text{"}0\text{"})),\,\dots .\,$$
The actual identity of
the other natural numbers is not important. What is of
great importance is that they are all definite successors
of zero. You are allowed to use any given set as the
set of natural numbers as long as one element is singled
out as the zero element and all the rest of the elements
are the successors of zero.
Thus, if you wish, you can use the set of even numbers as
a model of the natural numbers and then define
$\,2=S(0),\,$ $4=S(2),\,$ $6=S(4),\,$ $\dots.\,$
In this model the number denoted by $\,2\,$ is the successor of
zero but this does not change its properties in the model
of the natural numbers. For example, in this model, we
have $\,2\times 2=2\,$ using the Peano definition of $\,\times\,$ (multiplication) of natural numbers. This is because $\,2\,$
models the natural number "$1$" and has the same properties in the model as "$1$" has.
This is an example of the abstract nature of modern axiomatic mathematics. The natural numbers are not defined by what they
are, but by what they do. All models of the natural
numbers based on the Peano Axioms are equivalent in
the sense that they all have the same properties in the model.
The Wikipedia article Peano axioms
has a lot of details, but the fundamental idea is that it is an axiomatic model of the natural numbers. Previously generations of mathematicians took them as given somewhat as in the quote
"God made the integers, all else is the work of man". The natural
numbers have not changed, but our view of their nature has.
Best Answer
The fact that every number is either 0 or a successor follows almost embarrassingly quickly from induction on the predicate $$P(x) \equiv (x = 0 ) \lor (\exists y)[x = S(y)].$$
Clearly $P(0)$ holds. Also $P(S(y))$ holds for all $y$, so $(\forall y)[P(y) \to P(S(y))]$ also holds. Now apply induction.
As you can see, the only axiom that is required here is the induction axiom for $P(x)$, along with usual first-order logic. The numerical axioms of $PA^{-}$ are entirely irrelevant.