Proof of internal resistance acting like it does I believe Ohm's law via the drude model (about average collisions time).
Right after, I read about internal resistance of a battery, and they used Ohm's law as a justification.
That doesn't make sense though; in a battery the current flows (in the battery) from MINUS to PLUS, which means Ohm's law should not give you any info on the current.
Is there an explanation (And in fact definition) of what we mean by a battery, and thus why there is a constant of internal resistence?
 A: Internal resistance of a battery is a bit of a "catch all" phrase.  There is a chemical reaction going on in a battery, whereby electrons flow out of the anode and into the cathode, because positively charged ions must flow to one terminal of the battery while negatively charged ions must flow to the other terminal of the battery, in order to provide the electrons to the cathode that carry out that chemical reaction (see https://www.upsbatterycenter.com/blog/battery-cathode/).  Those ions have to travel through an electrolyte, and there is some distance that they must travel and quite a few molecules that those ions must pass, in order to get to their target terminal.  Since those ions can't flow at infinite velocity, there is a limit to how much current you can get out of any battery, even if you short circuit the terminals of that battery.
A: Internal resistance is just a concept for approximating the non ideal behaviour of a battery. This latter is ideally defined as a device that is capable of provinding a constant voltage regardless of the current the circuit being fed is draining. If you were to plot such a device in a V(I) graph, it would be a horizontal line.
However, such a device does not exist in reality, meaning that the voltage provided will be constant only for a certain range of drained current (and many times not even for that): as a first approximation one could model this non-ideality as a line with negative slope in the V(I) graph: the more the current required, the less the actual voltage supplied. But such a line describes a negative resistor from the perspective of the generator, or a positive resistor from the fed circuit perspective: this artificial resistor is called internal resistor.
It describes very well a series of actual phenomenon of a battery, such as joule heating, for a certain operative range (I), but there is no resistor component as such inside the battery.
A: It's really just calculus. In any situation where voltage, E, is dependent on current, I (or vice-versa), you may replace E/I=R with dE/dI=R(I). Then, for a given test current you may state a resistance. The actual physics behind R(I) in batteries is complex, as the other answers point out. In practice, if you want to know R(I), you need to measure it.
