Does the definition of rational numbers already assumes an understanding of them? I mean, how do we know that the correct equivalence relation is the equality of the crossed products? Does it already assume an intuitive informal understanding of rationals? If yes, what would that intuition be?

What do you mean "correct" equivalence relation?
In a way the answer to your question is "yes" because or goal is to take our intuitive understanding of rations and formalize it in a way so that our formulation fits our intuition.  But in a more important way the answer to your question is "no", because our goal is that are formalization does not rely on any intuition and can apply to other mathematical structures that may prove very different than our intuition.
Our formulation is the $\mathbb Q$ is the set of equivalence classs on the relation on $\mathbb Z\times \mathbb N$ that $(a,b)\sim (c,d) \iff ad=bc$.  And for that formulation all we need to prove is that $\sim$ is an equivalence relation which is fairly straightforward to show.
There is no question that it is the "correct" equivalence relation.  Correct for what?
We can show that if we do have an intuitive idea of what that rationals are that this equivalence fits our intuition.  If we intuitively have some idea that $\frac ab$ is $a\div b$ and $\div$ is the inverse of $\times$ then if $a\div b = c\div d$ then $(a\div b)\times b = (c\div d)\times b$ so $a = (c\div d)\times b$ so $a\times d =(c\div d)\times b \times d = [(c\div d)\times d]\times b= c \times d$.
And alternatively if $ad = bc$ and $d\ne 0;c\ne 0$ then $(ad)\div d = (bc)\div d$ so $a =(bc)\div d$ so $a\div c = [(bc)\div d]\div c= [(bc\div c)]\div d = b\div d$.
So yes, it fits our intuition.
