The two forces have equal status. You cannot consider one of them to be "cause" and the other to be "effect." You should not take the words "action" and "reaction" in the context of the Third Law as indicating "cause" and "effect". They are simply a commonly-used terminology.
[added] When I teach Newton's Third Law in an introductory course, I avoid using the words "action" and "reaction" except to address confusions such as this.
I like your explanation, and I agree. However, why does it not work for the case of gravity? To be more specific, I'm talking about the mainstream classical justification for the equivalence principle as it applies to active gravitational mass. Let me give an analogy that applies to the OPs question:
Two men with a rope are standing face to face on a frictionless surface (or ice). Regardless of which man pulls on the rope, one, the other, or both, they will always meet at their center of mass. Just as your post describes, the third law does not discriminate between the action and reaction.
However, it does discriminate for the case of gravity:
In fact, the equivalence principle depends on the discrimination! In the two body problem, if the gravitational field of M1 disappears, then the third law will be violated (they will not meet at their center of mass). And this is used as the justification for the equivalence of active gravitational mass and passive gravitational mass, even though there is no physical evidence to support it.
Am I missing something?