There are many ways to define the complex numbers. Each such definition should consist of the following:
- A "list" of all complex numbers. Two examples are the formal expressions $x+yi$ for real $x, y$ (or, which is the same, pairs $(x, y) $), and polar representation, i.e. $(r,\theta) $ for real $r> 0$ and $\theta$ plus the number $0$. Each complex number should have a unique representation. Alternatively, we can provide a rule for deciding when two complex numbers are equal, but that complicates the further steps of the construction.
- An embedding of the real numbers into the complex numbers. In other words, for each real number there should correspond some unique complex number.
- A definition of the field operations of addition and multiplication. These are easier in the non-polar representation.
- An identification of one of the roots of $-1$ as $i$ (the field operations don't distinguish $i$ from $-i$).
Given these, one should be able to derive the usual representations of the complex numbers, and from these define the function $\exp z$; this is a definition. One way of defining this function is through the formula you mention, and another is using an infinite series (this requires more work since you have to define limits).
More abstract definitions are possible. For example, we can define the complex numbers as the algebraic closure of the reals (the smallest field containing the reals in which every polynomial has a root).
Using the uniqueness of the algebraic closure, one can show that all definitions of the complex numbers are the same, that is they result in the same notion of complex numbers. In fact, suppose we have a field extending the complex numbers, possessing a root $i$ of $-1$, such that every number can be written in the form $x+yi$; then this field is isomorphic to the complex numbers.
Summarizing, there are many ways of defining the complex numbers, and they all result in the same notion of complex numbers. The exponential function is a definition and doesn't form a part of the construction. Other derived concepts are the complex conjugate, the norm, and convergence.
The difference between essential and derived notions is more philosophical than practical. You might as well say that the derived notions form part of the construction. This is just another point of view, which perhaps makes sense from the perspective of formal logic.