The terminology “toric structure”, “toric line bundle” and “toric section” comes from the fact that the total space of a toric line bundle $V(L)=\operatorname{\bf Spec}_{X}(\operatorname{Sym}(L^{\vee}))$ admits a unique structure of toric variety satisfying the conditions:

1. (1)

   $z$ is the distinguished point of the principal open subset;

2. (2)

   the structural morphism $V(L)\to X$ is a toric morphism;

3. (3)

   for each point $x\in X$ and vector $w\in L_{x}$, the morphism $\mathbb{G}_{m}\to V(L)$, given by scalar multiplication $\lambda\mapsto\lambda w$, is equivariant;

4. (4)

   every toric section $s$ determines a toric morphism $U\to V(L)$, where $U$ is the invariant open subset of regular points of $s$.

This can be shown using the construction of $V(L)$ as a toric variety in [Oda88, Proposition 2.1].