Notations
1.$\Sigma_g$ is the Reimann surface with genus $g$
2.$M_g$ is the space of all metrics
3.Diff($\Sigma_g$) is the diffeomorphism on $\Sigma_g$
4.$\text{Diff}_0(\Sigma)$ is the connected component of the identity of Diff($\Sigma_g$)
5.Weyl($\Sigma_g$) is the Weyl transfromation on $\Sigma_g$
Contexts
On the page 470 in Nakahara, the author defines the Moduli space as : $$M_g / \text{Diff}(\Sigma_g)\times \text{Weyl}(\Sigma_g)$$
Teichmüller space as : $$\text{Teich}(\Sigma_g) = M_g / \text{Diff}_0(\Sigma_g)\times \text{Weyl}(\Sigma_g) $$
and Modular Group(MG) or Mapping Class Group (MCG) as : $$\text{Diff}(\Sigma_g)/ \text{Diff}_0(\Sigma_g) $$
However, as one discusses the torus, there seems to be different definitions of these terminologies. (If I understand correctly )One can define the atlas of the torus by the equivalence relation on $\mathbb{C}$ :
$$z \sim z + n \omega_1 + m \omega_2 \quad ,\quad m,n \in \mathbb{Z}$$
Define $\tau \equiv \frac{\omega_2}{\omega_1}$. In Nakahara p.270, the modular transformation is generated by $\tau \rightarrow \tau +1$ and $\tau \rightarrow -\frac{1}{\tau}$. The torus with the same complex structure can be related by modular transformation. Besides, in Gleb Arutyunov - String lecture note p.90, $\tau$ is the Teichmüller parameter and the region $\text{Im} \tau > 0$ is the Teichmüller space.
Questions
Are these two kinds of definitions (MG,Teich space and modular transformation)equivalent? Why? It seems to me that they are totally different since the former involves the metric structure; nevertheless, the latter doesn't. However, they seems to be exactly the same in the case of torus.
I'm not sure I've ever seen the Teichmuller space and the modular space defined directly using the full space of metrics modulo Weyl transformations; perhaps that's a physics thing.
Instead, the usual practice in geometric topology and geometric analysis is to apply the Uniformization Theorem, which says that every conformal class of metrics has a unique representative which is normalized in an appropriate manner: when $g \ge 2$ this means that the sectional curvature is $-1$; whereas when $g=1$ this means that the sectional curvature is $0$, and one special geodesic loop on the torus has length equal to $1$. One side effect is that taking the quotient by Weyl transformations is entirely unnecessary.
It just so happens that in the case that $g=1$ a special thing happens: the space of normalized metrics may be defined as you say using lattices in $\mathbb C$. For $g \ge 2$, there's no such special thing happening.