Is "force" and "exerting a force" the same or different things? Is "exerting a force" and "force" the same or different things?
Either it your answer is "same" or "different" give an explanation for this.
How this doubt came to my mind?
Actually all of the definitions I can see are describing force as an action like one of the most famous definiton - "a force is a push or pull". Push and pull are both actions which means that force is also an action. But in my mind it also feels like that force is just a magnitude for rate of change in momentum.
 A: "There is a force on car" - here force is showed as an action.
Force is an action but we need to measure that action and that measure of action is called the amount of force or simply  force.
A: The precise definition of force is the equation $\vec F = {d \vec p \over dt}$ where $\vec F$ is net force and $\vec p$ is linear momentum.  Linguistic descriptions, such as exerting a force or force, provide word-based descriptions but are not precise definitions.
As you say, the best linguistic description of force is stating the equation using words: force is the rate of change of momentum; but even this description does not capture the vector nature of force as does the equation.
A: According to Aristotle, force is that which causes change and this by contact and hence force has to be exerted. Aristotle got the general law correct, but the particular application of this to mechanics wrong.
This is manifest in Newton's second law of motion which states that:

force = rate of change of momentum.

In Aristotelian terms, force causes a change of momentum. Force is hence causal.
T'hooft, by the way, considers Newtons second law to be causal in the way that Aristotle phrases it, even though the traditional formulation of it is acausal.
I also point out that given Aristotles definition of force, then both the effect of spacetime on mass-energy and that of mass-energy on the metric are forces since they both cause change. Since physics is about change, one can say that the universe is a network of forces. But then of course the work comes in distinguishing different types of forces
A: 
Is "exerting a force" and "force" the same or different things?

I assume you are asking if the mechanisms involved for exerting or experiencing a force are the same thing. While a force is a force, the mechanisms for exerting or experiencing a force may be different.
Let's say I push on your chest with my hand with a force of magnitude $F$. Per Newton's third law your chest applies an equal and opposite force of $F$ on my hand. The force I applied to you was a force exerted (actively applied) to your chest, an "action force" if you will. But the force your chest exerted on my hand is a reaction to the force I applied to you, a "reaction force" of you will. You did not purposely use your chest to apply a force to my hand, while I purposely applied a force to your chest.
Now when it comes to the effect of the action and reaction forces on each of us, we need to turn to Newton's second law. That law tells us that the potential acceleration each of us may experience as a result of the action/reaction pairs of forces will depend on the net force applied to each of us individually. For that you need to include other external forces acting on each of us, such as the static friction force between our feet and the ground which opposes the action-reaction forces applied to each other.
Hope this helps.
