  We prove that the canonical model of a $3$-fold of general type with
geometric genus $2$ and with minimal canonical volume $\frac{1}{3}$ must be a
hypersurface of degree $16$ in $\mathbb{P}(1,1,2,3,8)$, which gives an explicit
description of its canonical ring. This implies that the coarse moduli space
$\mathcal{M}_{\frac{1}{3}, 2}$, parametrizing all canonical $3$-folds with
canonical volume $\frac{1}{3}$ and geometric genus $2$, is an irreducible
variety of dimension $189$. Parallel studies show that $\mathcal{M}_{1, 3}$ is
irreducible as well and is of dimension $236$, and that $\mathcal{M}_{2, 4}$ is
irreducible and is of dimension $270$. As being conceived, every member in
these 3 families is simply connected. Additionally, our method yields the
expected Noether inequality $\textrm{Vol}>\frac{4}{3}p_g-\frac{10}{3}$ for
$3$-folds of general type with $p_g=5$.
