To give one answer to the question in the title: A reason to prefer one way of defining the same 'idea' over another is that it generalizes better. (Where of course what better means can depend.)
I have notthing to say about equivalence relations but since also other examples are asked for:
The notion 'prime number' can perhaps serve as an example for what I mean.
For the natural numbers one can define this in various ways equivalent ways. In particular (we exclude $1$, and $0$ if one considers it as natural number):
A number $p$ is a prime number if $p=ab$ implies $p=a$ or $p=b$. (Or, put differently, $p$ is only divisible by $1$ and itself)
A number $p$ is a prime number if $p|ab$ implies $p|a$ or $p|b$.
These are equivalent for the natural numbers. But, I am convinced that the latter is the better definition of prime number. (Though, to answer side questions, when teaching introductory things I might use the former, since students might be already familiar with this being the definition and anyway I will not have the time to convey why the other is better. Also, I think this conviction is not universal (now) and certainly was not earlier.)
So, why am I convinced about this. Because, if you part from the natural number/integers to more general things, say rings of algebraic integers, Dedekind domains, domains in general, it is this definition that generalizes better to yield a notion of prime element.
The property given by the former is also interesting in more general situations, but I/one(?) would not call such an element prime but rather irreducible, for example.
Thus, in this case I would not say there are two definition for the 'idea' prime number that are equivalent.
But, rather there are really two 'ideas', the one of an irreducible element and the other of a prime element, and for the natural numbers (and the integers) one can show that they coincide.
The former definition yields the former, and the latter the latter. And, the latter being somehow more pertinent, as also documented by naming, this is the definition of prime number (in the naturals) that I prefer, since it generalizes better.