Mathematically, by relating the intrinsic quantities like drift velocity,relaxation time etc., it is easy to show that




are true.

But, is it correct to say the the first equation is really a generalisation of the second equation? If yes why?


$\rho$ is a property of a material, $R$ is a property of a resistor or some other specific object made out of some material or a combination of materials. They are related, but neither one is a generalization of the other.

A close analogy is density and mass. They are related, but you cannot say that a mass is a special kind of a density or density is special kind of a mass.


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