Charge densities when placing a conductor or a dielectric material inside a capacitor

The following might be a useful approach to consider:

When dealing with conductors, one key condition that must be satisfied is that the net electric field inside a conductor is zero. The charges in the conductor rearrange themselves to make this happen and furthermore, they will be located on the surface of the conductor.

Taking the case of two infinite parallel conductors with surface charge densities $+sigma$ and $-sigma$. The below figure shows the idea. The E-fields from each surface are shown with their relative magnitudes and direction. Note that the field between the conductors has a contribution from both the upper (+) conductor, and the lower (-) conductor. The E-field in the middle region is proportional to $sigma$ and points from the top conductor to the bottom.

Further note that the field inside both conductors is zero. Again, this condition is made possible by contributions from both conductors.

Now, consider adding a conductor between the two plates. The below figure shows the situation. Note that the charges in the newly added conductor have equal positive and negative contributions, since it was assumed neutral to begin with. However, the charges distribute in such a way as to make the E-field inside zero. Note that this time, the cancellation is due to contributions from the upper and lower surfaces of the same conductor. This cancelling field from the middle conductor zeros out the E-field from the original two plates within the middle conductor itself, but doesn't change the E-fields in the other regions.

I hope this helps.