We know there are several definitions about forcing equivalence which imply that two forcings notions can be equivalent or not. In general we like to know the similarity between generic extensions by different forcings. Among them, we are familiar with complete embedding, densely embedding and so on. As a matter of fact If $\mathbb P$ and $\mathbb Q$ are two forcing notions, let's call them equivalent if their Boolean completion are isomorphic as Boolean algebras, then if it is the case we have V[G]=V[H], for any $\mathbb P$- generic filter $G$ and $\mathbb{Q}$- generic filter $H$.

I am interested to understand similarity between satisfaction of generic extension by different forcing notions forcing notions, let's unpack this:

Definition: Fix $V\models ZFC$, Let $P$ and $Q$ be two forcing notions in $V$, we say $\mathbb P$ is weakly equivalent to $\mathbb Q$, if $V[G]\equiv V[H]$(i.e $V[G]$ is elementary equivalnet to $V[H]$), for all $G$ and $H$ which are $\mathbb {P}$- generic and $\mathbb Q$- genric filters over V, respectively.

Remark 1: Weakly equivalence probably is not definable in $V$.

Question: Is there any characterization of the above-mentioned notion based on Boolean algebras or order structure of forcings? Or at least sufficient conditions for being weakly equivalent and there is nontrivial example i.e two foricngs which are not equivalent but weakly equivalent.

Remark 2: By a theorem of Golshani and Mitchell, there is a model of $ZFC$, such that $Coll(\omega, \kappa)$ is weakly equivalent to $Coll(\omega,\lambda)$, for all $\kappa$ and $\lambda$ infinite. Thus consistently there are forcing notions which are weakly equivalent, but not equivalent.

Remark 3: Note that in Golshani-Mitchell model, trivial forcing is also weakly equivalent to collapse forcing, so it seems the situation is complicated.