Question about the Derivation of the cable equation for neurites
The part you are quoting is about helping you conceptualize the cable equation by starting from some approximations that let you remove most of the terms. See the sentence beginning (emphasis mine):


  To better understand how the cable equation is derived, first simplify the theoretical neuron even further and pretend...


They then walk you through how, because these approximations are just pretend, one needs to include terms to account for the actual permeability and capacitance of the membrane. This means that you do not have a constant internal current with distance as you travel along; instead, the current decays with distance because of leak through the membrane and membrane capacitance.

This is what the last equation you quote from says: the change in internal current over distance x (written as dil/dx) is equal to the current lost to the membrane, which includes two parts: leak through the membrane (V/rm) and charging the membrane capacitor (cm*dV/dt).



Another correction that might be tripping you up: you write


  change in voltage over time


in your question, but that is not what the equations are representing, they are representing change in voltage over distance x.