I was wondering which interpretation could we find for the resistivity, what image correspond to the concept.
Furthermore, how to interpret its unit $\Omega \:\rm m$ . Why is it more logical than a $ \Omega /m$ ? (I'm not seeking a proof by formula but a physical sense)
Fifty or sixty years ago, it was quite common for textbooks to define the resistivity of a material as the resistance between opposite faces of a unit cube of the material. Ignoring units, that is literally correct, and it may or may not be of help in grasping the concept. But this so-called definition has a number of faults…
To measure the resistance between opposite faces of a $1\text{m}^3$ cube, it would be no good touching your ohm-meter probes on to points in the middle of the faces. You'd need to make contact across the whole of these faces. That means you'd need connecting leads of cross-section greater than that of a cube face. In other words you can't measure the required resistance!
The 'definition' based on a unit cube gives you no clue as to how resistivity enables you to find the resistance of a conductor of any other size or shape. Indeed you might be misled into thinking that the greater the volume of your conductor, the greater, necessarily, its resistance. So the modern definition is as the constant $\rho$ in$$R=\rho\frac{L}{A}.$$in which $R$ is the resistance of a wire of length$L$ and cross-sectional area $A$.
This equation, having made $\rho$ the subject, gives the units of resistivity as $\Omega\ \text{m}^2/\text{m} = {\Omega\ \text m}$. The unit cube gives them, misleadingly, as $\Omega$. I can't think of any argument, even a plausible invalid one (I don't regard ignoring the $A$ dependency as plausible), that yields $\Omega\ \text{m}^{-1}$ as the unit.
When we first meet $\Omega\ \text m$ it does seem weird. The reason is that the resistance of a wire has different dependencies on its length and cross-sectional area; the relationship is not as simple as a dependency, say, just on volume or just on length, such as one might have met previously, in other contexts.
To sum up, I'd urge you to ponder the role of $\rho$ in the equation above. It's the factor that takes account of the effect of the material (at a particular temperature) on the resistance of a conductor. [The factor $L/A$ can be understood by regarding the wire as a number, proportional to $L$, of identical smaller lengths in series, each consisting of a number, proportional to $A$, of identical strands in parallel.]
I assume you have an understanding of what resistance is.
Let's say you have some wire. It has a certain electrical resistance (measured in $\Omega$). If you have an identical wire that is twice as long, it will have twice the resistance. If you have an identical wire that has twice the cross section it will have half the resistance of the original wire.
Now, physicists are usually not interested in a particular wire, but in the general case. If you are going to make a theory for current through a metal wire, you are not going to make that for say a $1~m\times 1~mm^2$ wire, because that theory would be somewhat narrow, limited to one specific case. (What if my wire is 1.5 m long?)
In order to get a more general theory you get rid of the length and cross section dependence, simply by studying the resistivity, $\rho$, which is defined through the equation
$$R=\rho\frac{l}{A}$$
By this procedure you get rid of the size dependence and $\rho$ will depend basically on the material only (and temperature, etc). Also it explains the unit $\Omega m$.
A procedure like this is very common in physics and usually denoted by the word specific, i.e. you could call $\rho$ the specific (electrical) resistance.
Another example from physics is the "specific heat capacity", which is the heat capacity per kg. Just like the resistivity, the specific heat capacity depends on the material only unlike the heat capacity which is proportional to the mass.
Or an example from everyday life: If you go to a shop to buy 0.5 kg of apples you pay a price for them. If your friend goes to the same shop to buy 2 kg of apples he will pay a different price (four times as much as you), simply because he bought a different amount of apples. In order to avoid having to display prices for 0.5 kg, 2 kg, and all kind of other amounts, the shop just displays the price per kilogram (in physics language, the "specific price" of apples). So, while there is some interest in the price of 2 kg/0.5 kg of apples in certain situation (namely for you and your friend), for the general case, the "specific price" is much more convenient.
To answer your question, whether to use resistance or resistivity is basically just a question of context. If you are interested in a particular piece of resistor, you are likely concerned with its resistance. If you are studying more general things, resistivity is your quantity of choice.
