The moduli space $\overline{M}_{1,2}$ can not be the blow up of $\mathbb{P}^{2}$ in $2$ points because its rational Picard group has rank $2$, indeed it is generated by the divisor parametrizing genus $0$ irreducible nodal curves with $2$ marked points and the divisor parametrizing reducible curves whose components are a smooth genus $1$ curve and a smooth genus $0$ curve with $2$ marked points.

The moduli space of genus $1$ stable curves with $2$ marked points is a rational surface with four singular points. Two singular points lie in $M_{1,2}$, and are:

a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve of Weierstrass representation $C_{4}$ with marked points $[0:1:0]$ and $[0:0:1]$;

a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve of Weierstrass representation $C_{6}$ with marked points $[0:1:0]$ and $[0:1:1]$.

The remaining two singular points lie on the boundary divisor $\Delta_{0,2}$, and are:

a singularity of type $\frac{1}{6}(2,4)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C_{6}$ and a smooth rational curve connected by a node;

a singularity of type $\frac{1}{4}(2,6)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C_{4}$ and a smooth rational curve connected by a node.

From this one can prove that:

**The moduli space of genus $1$ stable curves with $2$ marked points is isomorphic to a weighted blow up of the weighted projective plane $\mathbb{P}(1,2,3)$ in its smooth point $[1:0:0]$. In particular $\overline{M}_{1,2}$ is a toric variety.**