Weaker version of locally extended residual finiteness 
Definition (Property A). We will say that group $G$ has property A if for every finitely generated subgroup $H$ of $G$ there exists a group morphism $\rho : G \to F$ to a finite group such that
  $\rho(H) \neq \rho(G)$.

Of course, property A implies residual finiteness but seems to be a stronger property. (Edited later: As pointed out by YCor this is, of course, false! )
Property A is related to the concept of LERF groups but seems to be a weaker property. Recall that a group $G$ is called LERF if for every finitely generated subgroup $H$ of $G$ and every $g \in G \setminus H$ there exists a finite index normal subgroup $K$ containing $H$ but not containing $g$.

Question 1. Has property A already appeared in the literature ? Is it actually weaker then locally extended residual finiteness ? 

My second question concerns finitely generated subgroups of matrices groups.

Question 2. Does there exist a finitely generated subgroup $G$ of $GL_n(\mathbb C)$ (for some $n$) which does not have property $A$ ?

 A: A restatement is that for every f.g. subgroup $H\neq G$ of $G$ there exists a normal finite index subgroup $N$ of $G$ such that $HN\neq G$. So this is also equivalent to the condition that every proper finitely generated subgroup of $G$ is contained in a proper finite index subgroup. This appears in the literature as the "engulfing property". Its hereditary version (groups in which every subgroup of finite index has the engulfing property) is considered here (esp. Section 3.2) as "Property LPF". Many examples are described in this link.
Your claim that this property "of course implies" residual finiteness (RF) is not correct. Indeed, (same link), f.g. virtually nilpotent-by-polycyclic satisfy the even stronger property PF that every proper subgroup is contained in a proper finite index subgroup. This includes non-RF groups.
To answer your first question, the converse also fails. For instance, by Margulis-Soifer + Weisfeiler, the RF group $\mathrm{SL}_k(\mathbf{Z})$ does not have Property LPF for $k\ge 3$ (hence some - possibly all- of its finite index subgroup fails to have the engulfing property). This also answers your second question.
As you observe, the engulfing property (and actually LPF) is implied by LERF, as discussed in the above link. The converse indeed fails, for instance the Baumslag-Solitar group $\mathbf{Z}[1/n]\rtimes_n\mathbf{Z}$ is LPF but not LERF for $n\ge 2$. This answers the last part of your first question.

Edit: I've been wondering about the difference between the engulfing property EP and property LPF (which means that every finite index subgroup has the engulfing property). I'm unable to produce an example with EP and without LPF (i.e. showing that EP does not pass to finite index subgroups), still I believe it might exist. If I'm not wrong, all papers proving EP for some class C of groups (usually related to geometric topology in some way) actually prove LPF, typically because the concerned classes C are stable under taking finite index subgroups.
