Resistance and resistivity: which one is the intrinsic and which is the geometric property? Why?

Resistivity is an intensive property. Resistance is neither extensive or intensive. However for the simple case of resistors wired in series, the resistance behaves extensively. Resistivity is regarded as a material property (1 over the dc conductivity), while resistance is a cumulative or bulk effect. If you had a wire say, whose resistivity varied along its length, you would need to integrate this along the length of the wire to get the resistance. Or, put another way, the resistance of a wire increases with length, while the resistivity does not (assuming a homogeneous wire).

The first formula you quote from wikipedia, is for a specific model: "uniform cross section with a uniform flow of electric current, and are made of one material." So not surprisingly if you measure the resistance and assume the resistivity is constant, then you can get average resistivity as shown. But this is not a general result.

The resistivity is a material property, thus it's intrinsic. Depending on the shape of the piece of the material you are using, the resistance will change accordingly.

Resistivity is the resistance of a given material when the material is of unit length and unit area. So, resistivity is an intrinsic property. Resistance changes with the material geometry, for example, the resistance of the material is doubled when length of the material is doubled and halved when area of cross-section of the material is doubled. However, in both the above cases, the resistivity of the material remains the same because it is still calculated over a unit length and a unit area of the material.