When does the choice of the generic matter? It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases where we force below some sort of master condition, for example, but this basically amounts to asking my question for the cone below the condition). This position might possibly be preferred by mathematicians who interpret talking about generic filters as "semantic sugar" for purely syntactic arguments about Boolean truth values, but if we are prepared to talk about generics as real objects, it seems weird that we don't pay very much attention to the structure of the objects which generate our extensions.
But perhaps we needn't always care about the particular generic. Let $M$ be a transitive model of set theory. Call a notion of forcing $\mathbb{P}\in M$ forcing agnostic (over $M$) if for any two $M$-generic filters $G,H\subseteq \mathbb{P}$ the two extensions $M[G]$ and $M[H]$ are elementarily equivalent.
There is an obvious reformulation of forcing agnosticism: $\mathbb{P}$ is forcing agnostic iff the Boolean value, with respect to the Boolean algebra associated to $\mathbb{P}$, of any sentence (without parameters in the forcing language) is either 0 or 1. This immediately implies that any almost homogeneous forcing is forcing agnostic; in fact, any two extensions by almost homogeneous forcing are elementarily equivalent in the language augmented with constants for elements of the ground model.
Is there a characterization of forcing agnostic posets? Is this a purely structural property of the poset or does the ambient model matter, i.e. can  a poset be forcing agnostic over some models but not over others?
I would also welcome any examples of forcing arguments where some care is needed in choosing the generic (in addition to ensuring a particular condition gets in).
 A: Here is another kind of example where the choice of the generic filter $g$ used to build a generic extension $M[g]$ of a transitive model $M$ is important.  Suppose that we have some set $x$ that we want to be an element of the generic extension $M[g]$.  If $x$ is not already in the ground model $M$, then a mutual genericity argument shows that for any condition $p$ there are lots of generic filters $g$ containing $p$ with $x \notin M[g]$.  So if we want $x \in M[g]$ then we must choose $g$ rather carefully.
For example, if $M$ is a countable mouse with a Woodin cardinal and $x$ is a real, then by Woodin's "genericity iteration" theorem (or a variant due to Neeman) there is an iteration $i: M \to M^*$ of $M$ and an $M^*$-generic filter $g$ such that $x \in M^*[g]$.  This theorem is used all the time in inner model theory, and there is no way to state it using only the forcing language of $M$ or of $M^*$.
A: Let me give a couple of examples to show at least that being forcing agnostic is not equivalent to being weakly homogeneous. 
First, start in any model of set theory $V$, and then add a Cohen real $c$. Work in $V[c]$. Let $\mathbb{P}=\mathbb{1}\oplus\text{Add}(\omega,1)$, the lottery sum of trivial forcing and the forcing to add another Cohen real. This is the forcing with an antichain of two elements, below the first it is trivial forcing, and below the second, it is the forcing to add a Cohen real. This is definitely not weakly homogeneous, since the condition that opt for the trivial forcing have a fundamentally different lower cone than the conditions that opt to add the Cohen real. But meanwhile, the extensions of $V[c]$ by $\mathbb{P}$ are either $V[c]$ itself or $V[c][d]$, where $d$ is another Cohen real, and all these models have the same theory as $V[c]$ itself, since the two-step generic $c*d$ can be thought of as one Cohen real, which leads to the same theory as $V[c]$. 
This example shows that the question of whether a given poset is forcing agnostic depends on the model in which it is considered, since clearly $\mathbb{P}$ considered in $L$ is not forcing agnostic, even though it is in $V[c]$, and this answers the question in your second-to-last paragraph.
Second, here is another kind of example. Consider a model with a tower of elementary substructures $V_{\kappa_0}\prec V_{\kappa_1}\prec V$, and let $\mathbb{P}$ be the forcing that either adds $\kappa_0$ many Cohen reals, or else adds $\kappa_1$ many Cohen reals (or you can do something else with the cardinals, such as collapse them, as long as individually the things are weakly homogeneous). The forcing $\mathbb{P}$ is not weakly homogeneous, since different conditions treat $\kappa_0$ and $\kappa_1$ differently, but this is not revealed in the theory that is forced by those conditions, because of our assumption that $V_{\kappa_0}\prec V_{\kappa_1}\prec V$. 
Third, another kind of example can arise when one performs forcing over an uncountable  model $M\subset M[G]$, in such a way that there are more than continuum many (in the meta-theory) intermediate models $M[G_\alpha]$. Thus, at least two of them will have the same theory, and so we will have $M[G_\alpha]\subset M[G_\beta]$ being agnostic, even though the iteration could be rather arbitrary. We cannot necessarily restrict to the cone of conditions forcing this specific theory, however, since perhaps no single condition determines the whole theory (see comment below).
Lastly, let me say that there are a number of arguments where one must choose the generic very carefully. 


*

*In the indestructibility arguments using the lottery preparation, one must work below a condition in the $j(\mathbb{P})$ forcing that opts for the right forcing at stage $\kappa$. This is the analogue of the Laver function hitting the right poset at stage $\kappa$ in the Laver preparation. 

*A better example, which fulfills the request at the end of your post, arises in the case of forcing pointwise definable models. At the heart of this argument is a construction, due to Simpson, of building a very specific generic filter that in addition to being generic, ensures the pointwise definability property. There is no single condition that suffices (and pointwise definability is not an expressible property anyway, so we don't expect all the filters to have the property).  

*Another example of that nature arises in the proof that every countable model $M$ has two extensions $M[c]$ and $M[d]$ by adding Cohen reals, such that $M[c]$ and $M[d]$ have no common forcing extension. (Thus, the generic multiverse of a given countable model of set theory is not upward directed.) You can see an account of this argument and generalizations in Set-theoretic geology, but the specific argument has also arisen on MO here, here and here. The main part of the argument is a very careful construction of these generic filters. 
