Motivation for using $\sigma_{8}$?
The specific radius of $8 h^{−1} \text{Mpc}$ is used because the value of $\sigma_8$ turned out to be close to unity. To quote e.g. from Amendola &amp; Tsujikawa (2010), p. 39:


  If the cells have a radius of $8 h^{−1} \text{Mpc}$, it turns out that $\sigma_R$ is close to unity. Conventionally the normalization of the power spectrum is therefore given by quoting $\sigma_8$.