What is Binding energy, actually?

I think you know that two protons will repel each other, according to Coulomb's law. And the coulomb force of repulsion between two protons in a nucleus is approximately $250 N$ if they are separated by a distance of one fermi, which is very large considering the small mass of the proton.

So, as you might well know, Strong Nuclear force, for a short-range, manages to keep the nucleus together. Now, if you want to break the nucleus into its constituents, you need to do a certain amount of work- which is the binding energy.

Now, think about the nucleus before it was formed. All its constituents would have been separate entities. If you wished to make them into a single body, for the sake of spontaneity (my use of this word may not be entirely correct), the end product must possess lower energy than the sum of energies of independent entities.

This is rather counter-intuitive if you don't have an idea about Strong Nuclear forces, as there would be an increase in the energy of the system, by Coulomb's law.

But what actually happens is that the mass of the nucleus is less than the mass of its constituents taken one at a time. So, from Prof. Einstein's mass-energy equivalence relation, we can say that the energy contained in this mass was released during the formation of the Nucleus and this difference in mass is called the mass defect.

So, in order to break the Nucleus, you need to remove (or rather give back) what was holding them together-so, quite naturally, the binding energy is the same as the energy released from mass defect

EDIT

As a response to your edited question, I'm including an analogy. Let us assume that you have a ball initially placed on the top of a shelf of height $h$. Now, if it slips and falls down, it will lose its internal energy in the form of kinetic energy and you will have the relation $$mgh+\frac{1}{2}mv^2=mgh_{2}$$ (essentially a special case of third equation of motion) where $h_2$ is its present height. If you want to lift it back to the position it initially was at, you need to supply energy equal to the energy lost as kinetic energy. $$E=\frac{1}{2}mv^2$$ You might have noticed that we don't consider $GFP$ (Gravitational Force Potential) in the equation for energy that has to be supplied.

In a similar fashion, if the initial energy of the nucleus (before formation) is $E$, and the particles lose a mass $\Delta m$, then the energy of the nucleus will be $$E_{nucleus}=E_i-\Delta mc^2$$

Now to bring it back to the initial stage (The $E_i$ already includes the $NFP$ you're talking about, in the same way the $GFP$ is included in the analogy above ) Now, to take the nucleus back to its initial position, (with energy $E_i$) you need to supply it the binding energy. $$E_{nucleus}+E_b=E_i$$ rearranging, $$E_{nucleus}=E_i-E_b$$

Comparing this equation with that of formation of the nucleus, we have $$E_b=\Delta mc^2$$

As is stated in one of the texts above you start with a nucleus and then split the nucleus into individual neutrons and protons (the constituent parts of the nucleus).
The work done to split up the nucleus into its constituent parts is the binding energy of the nucleus.
In the reverse process if the individual neutrons and protons are brought together and form the nucleus the amount of energy released in that process is equal to the binding energy of the nucleus.

It is found that the mass of the nucleus is smaller that the total mass of the individual neutrons and protons which make up the nucleus.
The difference between these two masses is called the mass defect, ie the nucleus is deficient in some mass as compared with the sum of the masses of the individual particles which make up the nucleus.

If the binding energy is $E_{\rm b}$ nad the mass defect is $\Delta m$ then the two are related vis Einstein's equation $E_{\rm b} = \Delta m \,c^2$ where $c$ is the speed of light.

So to break up a nucleus into its constituent parts the minimum amount of energy input into the nucleus is the binding energy and at the end of the process the process the total mass of the constituent parts increase by an amount equal to the mass defect.