Electronic – Are the terms “Sheet Resistance” and “Sheet Resistivity” interchangeable

Just a note: the resistance between two points in space is highly dependent on the geometry of the space. Even things far away affect the electromagnetic properties of things close by, which means the approximation isn't very good except for wide open spaces.

Ok, so to get the resistance in the setting you mentioned you also need to factor the current emitter itself. It makes no sense to ask the resistance between two points if the full geometry of the problem isn't specified (and this includes the battery).

In practice (or is it theory?), this means you have to e.g. model your two points as charged spheres with a constant +I/-I into them and solve Maxwell's equations. If you shrank you spheres too much, the current would inevitably be concentrated in a too small area, and you get rapidly increasing resistance. In other words, the resistance between any two points assuming pointwise sources is infinite -- which shows mathematically that asking "what is the resistance of the air" is incomplete -- you need to specify everything.

Edit: I forgot to add that because of the linearity of Maxwells equations in simple media with uniform conductivity like the models I exemplified, the final resistance is going to be simply proportional to the conductivity (a simple subdivision argument will convince you of this), so you can compare different (linear/homogeneous/etc) conductors directly.

First, it's to emphasize that (in most cases) \$\alpha\$ << 1.

Secondly, \$\alpha\$ is a function of the material and temperature, it is not a function of the construction of the resistor. Any resistor of that material, at that temperature, would (ideally) have the same \$\alpha\$.

Your \$\alpha_2\$ is dependent on \$R_0\$ so it would be different for every resistor. That's very inconvenient.