Unfortunately, my question is a bit vague. Since I was not able to find anything about it in the literature, I decided to ask it on this forum. Here it is:
Let $A$ be a model in some language $L$. Let $L_A$ be the language containing $L$ and for each element in $A$ a constant naming this element. We may view $A$ as an $L_A$-structure with the obvious interpretation of the new constants. Assume that $\Sigma(x)$ and $\Pi(y)$ are types over $A$ (i.e. sets of formulas in the language $L_A$ which are finitely satisfiable in $A$). Call $\Sigma(x)$ and $\Pi(y)$ $\textbf{equivalent}$ over $A$ if for each elementary extension $B$ of $A$, the following holds:
$\phantom{aaaaxxxssssssssssaa}$$\Sigma(x)$ is realized in $B$$\phantom{aa}$iff$\phantom{aa}$$\Pi(y)$ is realized in $B$
As an example, the types
$$\Sigma(x)=\{x>n : n \in \mathbb{N}\}$$ $$\Pi(y)=\{r>y : r \in \mathbb{R^+}\} \cup \{y>0\}$$
are equivalent over the ordered field $\mathbb{R}$.
$\textbf{Question:}$ Is this concept of "equivalent types" studied somewhere in the literature on model theory (maybe under a different name)? Especially, I am interested in statements which are implied by or equivalent to the "equivalence of types". These statements may for example contain information about
$\bullet$ definability properties concerning the elements realizing the types or concerning the parameters from $A$ contained in the types
$\bullet$ automorphism properties of the models
$\bullet$ syntactical or logical relationships between the types and the formulas they contain
$\bullet$ topological properties of the stone spaces of the involved models
I should add that it will be very helpful for me if these questions are not only discussed for the case where the models in question (or their theories) are assumed to exhibit a certain degree of stability.