Why are the Resistance/Resistivity Equations Formulated as they are?
First, it's to emphasize that (in most cases) \$\alpha\$ &lt;&lt; 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.