I am Jinu Jeong.
In my opinion, there is no difference between minimization solved by energy and minimization solved by force.
Because the status of minimum energy is same with the status of zero net force.
By the way, the “min_style” command serves minimization style both measure.(energy, force)
In sum, my question is why min_style serves different minimization style.
If I understand your question correctly, you want to know why LAMMPS provides different minimisation algorithms? The reason is that different algorithms work better for different situations.
The default conjugate gradient style is typically a good balance between computational efficiency and stability, but there are probably situations in which it is better to use a more computationally intensive but more stable scheme, like a quasi-Newton method.
I suppose your first point it is true in theory, but in practice there might be subtle differences when dealing with truncated potentials.