How many ways can we define prime number?

Just like Wilson's theorem (a positive integer is a prime if and only if ....) any equivalent statement is handy when we want to show a number is prime. If we encountered a number in a situation where congruence condition of its factorial is available then it is good to use that for testing primality.

Logically any of the "if-and-only-if theorems" can be taken as a definition.

But human beings who discover theorems are not logical machines. While learning a subject, a formulation that uses less preliminary concepts is helpful; So the theorem "$n$ is a prime iff $Z/nZ$ is a field", is not suitable as a definition.

Accepted textbooks use that formulation of definition which makes it easy to digest and understand the subject, even if the subject did not evolve that way historically!

Take the statements of any of the deterministic pirmality tests. Each of their converse gives a new definition of primes.

I'm going to elaborate on the comment that a number $p$ is prime if $p \mid ab$ means either $p \mid a$ or $p \mid b$.

Consider for example $p = 14$. That's not actually prime, but indulge me for a minute. Check that $14 \mid 112$ and $112 = 7 \times 16$. However, $14 \nmid 7$ and $14 \nmid 16$ either. Therefore $14$ is not prime. However, $2$ and $7$ are prime, since for any choice of $a$ and $b$ such that $ab = 112$, you will see that either $a$ or $b$, maybe both, are divisible by $2$ and/or by $7$.

The definition you gave is equivalent to saying if $ab = p$, then either $a$ or $b$ is a unit (like $-1$). And that definition is fine until you venture out to domains of numbers other than $\mathbb{Z}$. For example, are the numbers $3 \pm 2 \sqrt{10}$ prime? Yes, they are. How about $\sqrt{10}$? It satisfies the $ab = p$ definition but not the $p \mid ab$ definition.