There are two problems to make this analogy precise:

1. If you think of site as presentations, they are presentation for a theory with operation of infinite arity (something like the theory of ininitary pretopos, i.e. categories satisfying all of Giraud axioms except being presentable, but including being co-complete).

Because of that you can expect to describe the transformation that you need to go from a site to an other by a finite sequence move involving a finite number of objects.

2. Site are presentations, but more in the sense of "generators + ideal", not "generators" + "generator of an ideal". So "adding definable relation" does not really makes sense, essentially any valid covering relation is already part of your topology. But you could consider the notion of topology generated by some class of covering of course.

Once this two problems acknowledged, Grothendieck Comparison lemma (see https://ncatlab.org/nlab/show/comparison+lemma) can be used to do something relatively close to what you have in mind:

It says something which can roughly be translated by "if I add a bunch of new object to my site that can be covered by the previous one then I don't change the topos of sheaves " (This is very rough, I refer you to the precise statement of theorem 3.2 on the nLab page).

Any equivalence between toposes of sheaves on two different sites can always be obtained by two application of this lemma (one going up and the other going down). Essentially, by taking a subcategory of the topos of sheaves large enough to contains the two sites that you have.

There are two observation to be made that limit a little this kind of analogy:

1. Site are indeed in some sense presentations, but an infinity theory (I mean with operation of infinite arity) something like the theory of ininitary pretopos, i.e. categories satisfying all of Giraud axioms except being presentable, but including being co-complete. Because of that you can expect to describe the transformation that you need to go from a site to an other by a finite sequence move involving a finite number of objects.

2. Site are presentations, but more in the sense of "generators + ideal", not "generators" + "generator of an ideal". So "adding definable relation" does not really makes sense, essentially any valid covering relation is already part of your topology. But you could consider the notion of topology generated by some class of covering of course.

Once this said, Grothendieck Comparison lemma (see https://ncatlab.org/nlab/show/comparison+lemma) can be used to do something relatively close to what you have in mind:

It says something which can roughly be translated by "if I add a bunch of new object to my site that can be covered by the previous one then I don't change the topos of sheaves " (This is very rough, I refer you to the precise statement of theorem 3.2 on the nLab page).

Any equivalence between toposes of sheaves on two different sites can always be obtained by two application of this lemma (one going up and the other going down). Essentially, by taking a subcategory of the topos of sheaves large enough to contains the two sites that you have.