What does it mean for the gravitational force to be "between" two bodies? What is the meaning of the word "between" in the law that the force between two masses at separation $r$ is given by $\frac{GM_1M_2}{r^2}$? I am confused about how can a force be in-between, either it is on body A or on body B, or on both.
Suppose body A exerts force $F$ on Body B, so according to Newton's 3rd law of motion B should also exert a force on A.
Let's consider this case for gravitational force between two bodies. If body A exerts force $g$ on Body B, then B body should also exert a force $g$ on A, but B is also exerting the gravitational force $X$ on A, hence A will also exert force $X$ on B.
So, how are two forces acting?
I have given the representation in this diagram.

 A: 
**If A body Exerts force G on Body B,then B body should also exert A force G on A,
but B is also exerting the gravitational force X on A,hence A will
Also exert Force X on B **

Like you said,  A experiences a force G towards  B , given by $\frac{GM_1M_2}{r^2}$.
B experiences the same force towards A.
These are the 2 forces in this scenario.
Where does X come into the picture. There is no other force X.
In your diagram " the reaction force of G " and X are not two distinct forces. They are the same thing.
Similarly, in your diagram " the reaction force of X " and G are the same thing
A: One way to look at this is that forces always come as pairs. For example, you start with a universe with only one object in it, then you add another object and nature will immediately create a pair of forces. It’s not like the Moon feels that the Earth is tugging at it, and retaliates by tugging at Earth itself.
You can’t take such a pair of forces and label one the action and the other the reaction, or one the cause and the other the effect. Rather, both are manifestations of inner workings of nature, and to the best of our current knowledge, those workings aren’t best described in terms of force, but rather as “if things were moving like this before, they’ll be moving like that afterwards”.
Despite not being fundamental, forces are very useful mathematical objects, and the symmetry they exhibit in Newton’s third law is but one among many symmetries the universe has.
A: The meaning of the word "between" in this case is the same as the meaning of between in the sentence:

The love between two people

Of course it is understood that love does not exist in air. One person loves another. Air has no brain and thus has neither emotions nor feelings.
The word between in the sentence above means that person A loves person B and person B loves person A.
Thus the phrase: "the force between two bodies" means body A exerts a force on body B (this force is on body B) and body B exerts a force on body A (this force is on body A).
A: It is not a individual force that exists in the space between them, it is rather saying between in the case that both bodies exert a force on each other, which gradually pulls them to a point between them as they are pulled towards each other, they ultimately simply are counterparts to each other, the equal and opposite reactions.
A: Newton's Third Law tells us that the force on A due to B is equal (in magnitude, with opposite direction) to the force on B due to A. Therefore, in any interaction between a pair of objects it is sufficient to describe the force on just one of them, since the other can be deduced by Newton's Third Law. For this reason, it is common to refer to force acting on either object simply as the force "between" the objects. Thus, there should only be one pair of forces, with magnitude $\frac{Gm_Am_B}{r^2}$, to describe the gravitational interaction between a pair of masses.
A: A (non-fictitious) force is a description of an interaction between 2 objects. So of course, the interaction acts upon both objects with equal magnitudes (Newton's 3rd Law). $A$ acts on $B$ with the same force as $B$ acts on $A$. Gravity is the same. It's an interaction between $A$ and $B$. You could separate it into 2 parts: force by $B$ on $A$ and force by $A$ on $B$. But really, they are 1 interaction.
Sidenote: Gravity as an interaction between $A$ and $B$ is only valid in the most basics of Newtonian mechanics. It gets described as other things later on, but I don't want to confuse you.
