The definition of potential energy is:- "The gravitational potential energy at a point in the gravitational field is defined as the work done to bring a particle from infinity to that point in the gravitational field. The work done referred here (the one I highlighted) is done by gravitational force or some other force?? I think it is done by gravitational force because in the calculations above we took force F = gravitational force. Kindly clarify.
Potential energy is calculated only for conservative forces. In this case the conservative force is the gravitational force.
Potential energy of a system is defined as negative of the work done by the internal conservative forces acting in that system.
$$dU = -dW_{c}$$ $$dU = -\vec{F}_{c}.d\vec{x}$$
Now there is an another way to calculate potential energy of a system. Consider an external force (not necessarily conservative) equal to the internal conservative force but opposite in direction of it. Under action of these forces, object is moved such that equilibrium is maintained at every point thus there is no change in kinetic energy of the system.
Then, from Work Energy Theorem:
$$W_{ext} + W_{con} = \Delta KE$$
Since, $\Delta KE = 0 $ therefore,
$$W_{ext} = -W_{con}$$
and $-W_{con} = \Delta U$ therefore,
$$W_{ext} = \Delta U$$
You can calculate gravitation potential energy from any of the two methods, but generally in textbooks the second one is mentioned.
Hope this helps
As your first equation indicates, the potential energy is equal to the work done to bring a particle from infinity to a point within the gravitational field. This work is accomplished by the gravitational field, not the force.
