Definition of $\exp(x)$

The following are equivalent definitions for $\exp(x)$. \begin{align} 1. & f(x) = \sum_{k=0}^{\infty} \dfrac{x^k}{k!}\\ 2. & \dfrac{d f(x)}{dx} = f(x) \text{ with } f(0) = 1\\ 3. & f(x) = \lim_{n \to \infty} \left(1+\dfrac{x}n\right)^n\\ 4. & f(x+y) = f(x) \cdot f(y) \text{ with }f(x) >0 \text{ being continuous at one point and } f(1) = e \end{align} If you start with any one, you can derive/prove the others.

EDIT

The important thing is that you can start with anyone and derive the others as property. If a statement $A$ implies a statement $B$ and vice-versa, both are equivalent statements. We may, hence, use any one of them as a definition.

For instance, if you choose $(1)$ to define $\exp(x)$ as $\exp(x) = \displaystyle \sum_{k=0}^{\infty} \dfrac{x^k}{k!}$, the rest from $(2)$ to $(4)$ become properties.

To address the comment below your question, and your assertion that something must have "one real" definition:

"Equivalent definitions" simply mean that there are many ways to define/describe $e^x$, all of which define/describe the unique function: $f(x) = e^x$. When definitions are equivalent, we can choose any one of them to derive the others. So we are free to choose which, among equivalent defintions, will serve our purpose best, depending on when and how we need to use it.

That is true of many mathematical entities: e.g., there is no *ONE* true definition of $\pi$: there are many ways to define the unique number $\pi$.

Another Wikipedia article of relevance is Characterizations of the exponential function.

Which characterization is most appropriate to take to be the definition depends on the context.