Bilinear transformation confusion
If you read the link which describes the second mapping
$$ w = \frac{z+1}{z-1}$$


  Thus inside of the unit circle in z-plane maps into the left half of w-plane and outside of the unit circle in z-plane maps into the right half of w-plane. Although w-plane seems to be similar to s-plane, quantitatively it is not same


It states that $w$ is not the $s$ plane.

Therefore, if your aim is in studying the bilinear transformation for digital filter design then you can safely use the first, well known, bilinear transformation relation, and its inverse:

$$\boxed{s \leftarrow \frac{2}{T}\frac{z-1}{z+1}}$$

Otherwise, the paper you linked is dealing with stability analysis of discrete-time systems using bilinear transform of its own selection...