Let $E\to X$ be a holomorphic vector bundle over a complex manifold $X$ and denote by $\mathbb{P}(E)$ its projectivization. There is then a notion of a tautological line bundle $\mathcal{O}_E(1)\to \mathbb{P}(E)$ (cf. Kobayashi section 3.5).
I do not understand why Demailly (in Complex Analytic and Differential Geometry page 281) considers $\mathcal{O}_E(1)$ to be the bundle $\pi^*E/S\to \mathbb{P}(E^*)$ where $S\subset \pi^*E$ is the tautological hyperplane bundle inside $E$.
I know these two bundles are related, but why does Demailly choose the less intuitive $\pi^*E/S$?