The proper way is to normalize it according to the weight of the electrode in order to get the specific capacitance in F/g units? or should I normalize it according to the geometric area to get it in F/cm^2?
I'm very familiar with that equation. The main question is what is the mass of the active material? the whole weight of the electrode? including the bulk of the electrode that probably is not active..
To answer your original question, my opinion is that researchers should report the values for both F.g-1 and F.cm-2. There are practical applications where one value may be more important than the other. You may have developed a new system that is fantastic for some applications, yet terrible for others. I feel researchers should be as transparent as possible.
The second question you asked is whether you should report the mass of the entire electrode. As an example, if I was using a glassy carbon electrode as a current collector, I would first determine the performance of this. I would then modify the surface with, for example, CNTs. I would then report the performance gain from adding the CNTs compared with the original surface. I do not think you can simply say "probably is not active".
Regardless, provide as many details as possible. There may be practical limitations, such as the fact a heavy/large current collector is required. I personally feel these things should not be overlooked.