I wonder if the definition of very ample sheaves of Hartshorne coincides with the definition of Stack projects.
Let $f: X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume that $X$ is noetherian so that Hartshorne's definition of "immersion" coincides with projects'.
Definition of Stack projects:
We say $\mathcal{L}$ is relatively very ample if there exists a quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$.
Definition of Hartshorne:
We say $\mathcal{L}$ is very ample relative to $S$ if there is an immersion $i: X \to \mathbf{P}_S^r$ for some $r$, such that $\mathcal{L} \cong i^*(\mathcal{O}(1))$.
If $\mathcal{E} = \mathcal{O}_Y^{\oplus r}$, then $$\mathbb{P}_Y(\mathcal{E}) = \operatorname{Proj} \mathcal{O}_Y[x_0, \dots, x_{r - 1}] = \mathbb{P}^{r - 1}_Y.$$
As such, Hartshorne's definition is a specialization of the stacks project's definition to the case that $\mathcal{E}$ is a free sheaf of modules.
Even when $\mathcal{E}$ is taken to be locally free, we still don't have an equivalence of the definitions, since according to Hartshorne's exercise II.7.14(a), there exists a noetherian scheme $Y$ and a locally free sheaf $\mathcal{E}$ on $Y$ so that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is not (Hartshorne) very ample relative to $Y$.