The Order of the Identity of a Group

The order of an element is a positive integer, by definition.

Don't think of $g^0$ as representing $g$ acting on itself $0$ times but as $g$ acting on something else $0$ times. Groups frequently represent actions on other objects. An important class of examples is symmetry groups.

Consider G as the group of symmetries of a square and $g \in G$ as rotate clockwise by $90^\circ$. Then $g^2$ is rotate clockwise by $180^\circ$ and $g^3$ is rotate clockwise by $270^\circ$. So, what is a sensible definition of $g^0$ in this context? Well, rotate by $0^\circ$. Similarly we can sensibly define $g^{-1}$ as rotate anticlockwise by $90^\circ$ since this will undo the effect of $g$. With these definitions, $g^n$ is rotate clockwise by $n \times 90^\circ$ whether $n$ is positive, zero, or negative and $g^m \times g^n = g^{m+n}$ and life is good. With your convention, things would not be so neat.

So, with this convention, $g^0$ is always $e$ and hence to be useful we must require the order to be $> 0$ and not just $\geq 0$.

You can define powers of a group element $g$: $g^0=e$ and $g^{n+1}=g^n\cdot g$ for $n\geq 0$. The order of an element is chosen to be $\geq 1$.