Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of constructing a segment congruent to an assigned segment”. His second axiom reads: "if two segments are congruent to a third one they are congruent to each other." Then he goes to prove the standard properties of equivalence relations follow from the axioms:
Since congruence or equality is introduced in geometry only through these axioms, it is by no means obvious that every segment is congruent to itself. However, this fact follows from the first two axioms on congruence if the segment AB is constructed on a ray so that it is congruent, say, to A'B' and Axiom III, 2 is applied to the congruences AB ≡ A'B', AB ≡ A'B'.
On the basis of this the symmetry and the transitivity of segment congruence can be established by an application of Axiom III, 2.


Now it is clear how we may define the general notion of equivalence relation. This "Hilbertian" defintion has at least two advantages: first, It avoids (at the start) the somehow non-intutive property of reflexivity; second, and more importantly, it is in more harmony with the standard way of defining an equivalence class consisting of everything equivalent to a focal element.


Here are my sub-questions:
Was the standard definition of equivalence relation just more lucky? Or, was there a rational choice involved? 
Do you know any other definition that has been preferred over the other? If yes, what was the reason?
Do you have a personal example of such "choice"? If yes, what is the reason of your preference? Do you have an "instructional" defintion of a certain concept that you prefer it over the "standard" definition of the same concept when teaching? if yes,... 



Why do mathematicians prefer one definition over the other when they both define the same concept?
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 nothing 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):


1. 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.)
2. A number $p$ is a prime number if $p\mid ab$ implies $p\mid a$ or $p\mid b$.


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.


