Why do we consider hydrostatic pressure and lithostatic pressure separately?
I think (but I'm not an expert on the topic) that your assumption is wrong! The pore fluid pressure and effective pressure exerted on the rock are not identical. And during log measurements, it is only the pore fluid pressure that is measured, $\sigma_\text{pore}=\rho_\text{water}gz$. As long as the water is connected through pores to the earth's surface and it is assumed to be hydrostatic (not-flowing), the measured pressure in the pores will create a pressure to push up the overlying water only -- the rocks will take care of carrying the overlying rocks themselves! If you'd drill a borehole, the water-filled column would be filled with water exactly to the earth's surface. The effective pressure on the whole system is then $\sigma_\text{effective}=\sigma_\text{lithostatic}-\sigma_\text{pore}$.

However, once a seal is formed (e.g., a shale formation overlies a formation), the pore water suddenly has to also carry the overlying rock formation! The pressure will then increase significantly, and the pore water pressure that you measure becomes more similar to the effective pressure. 

The pore water pressure thus increased below a seal because it is stuck. Once you drill a hole from the surface to this formation, you allow water to flow freely to the surface, and it is clear that it is under overpressure with respect to the lithostatic pressure, because it will blow out!

See also this comprehensive set of slides: http://ocw.utm.my/pluginfile.php/1509/mod_resource/content/0/To_upload_OCW_Formation_pressures.pdf .