A class K of L-structures is called an elementary class if there exists some theory T such that K is exactly the class of all structures M satisfying T.
Given a set Γ of first-order sentences over L, let Mod(Γ) denote the collection of L-structures modelling Γ. (eg — Γ is the group axioms, then Mod(Γ) are the groups.) The collection of elementary classes is the image of ϕ↦Mod({ϕ}). The collection of weakly elementary classes is the image of Γ↦Mod(Γ). (If a class is weakly elementary wrt finite Γ, then it is also elementary wrt the conjunction of Γ)
In other words, I think: an elementary class is an equivalence class of ≡. (todo: double-check. ah, this is wrong — this definiton is too strict. The class of all structures is elementary by the theory T=∅, but it is not equal to the equivalence class of the empty theory wrt ≡.) todo: apparently there are examples of elementary classes in the textbook — worth looking at