When can we find different axiomatizations for theories with the same models? The theory of groups can be axiomatized in the usual way, saying there must exist inverses, a unit element, and associativity must hold, or it could be written down in a "single axiom" as done here. This can also be done for Boolean algebras, ortholattices, and some other algebraic structures. It is worth noting that these axiomatizations seem to be in a language with just one binary relation, or just a binary relation and maybe a constant. Furthermore, the axioms themselves seem to be $\Pi^0_1$.
I should say, I'd describe the axiomatizations I am thinking about as "nontrivial" where a "trivial" example would be to just take the conjunction of the usual axioms for any given theory. Also, the simpler the language (so the signature), the more interesting I would find the axiomatization (e.g. a single axiom in a language with no constants is more interesting than one in a language with one constant). Also the simpler the axiomatization (e.g. no $\Sigma^0_2$ (or above) formulas), the more interesting.
So now I ask, when can we find alternate/alternate and interesting axiomatizations for theories that have the same models (so could be thought of as similar/the same theory)?
 A: One interesting example of this phenomenon, while far from the examples you give, is when we have an infinite-to-finite improvement: an infinite set of axioms $T$ which turns out to be finitely axiomatizable.
The two most common infinite systems of axioms are probably (modulo small variations) $\mathsf{ZFC}$ and $\mathsf{PA}$. It is known that neither theory is finitely axiomatizable, since in each case the theory proves the consistency of each of its finite subtheories - see here for the $\mathsf{ZFC}$ case. However, they do have fragments which still are most obviously infinitely axiomatized but now do turn out to have finite axiomatizations, specifically by restricting the relevant schemes (separation/replacement and induction, respectively) to a fixed level of an appropriate hierarchy (the Levy and arithmetical hierarchies, respectively). Finite axiomatizability of these "bounded" fragments follows from the existence of universal formulas in each class, like how the halting problem is a complete c.e. set. This "universal formula" trick gives finite axiomatizability for a lot of other theories too, e.g. each of the "Big Five" in reverse mathematics.
We even have a natural(ish) example which is not simply a fragment of a better-behaved theory; specifically, Quine's alternate set theory New Foundations and many of its variants.
