I am reading Javanpeykar's notes on Lang-Vojta's conjecture (https://arxiv.org/pdf/2002.11981.pdf), where I find a definition of "near integral points":
$\textbf{Def 1:}$ Let $X \to S$ be a morphism of schemes with $S$ integral. Define the near $S$ integral points to be the set of points $P$ in $X(K(S))$ such that, for every point $s \in S$ of codimension $1$, the point $P$ lies in the image of $X(\mathcal{O}_{S,s}) \to X(K(S))$.
He also cite the Vojta's paper (https://link.springer.com/chapter/10.1007/978-3-319-11523-8_8), where also have a definition of "near integral points":
$\textbf{Def 2:}$ Let $B$ be an extension of an entire ring $A$. Let $\mathcal{X}$ be a model of $X$ over $A$. A near $B$ integral points of $X$ relative to $\mathcal{X}$ is a rational map $\text{Spec} B \dashrightarrow \mathcal{X}$ over $\text{Spec} A$, regular outside a subset of $\text{Spec} B$ of at least codimension $2$.
If we take $S$ in $\textbf{Def 1}$ to be $\text{Spec}B$, I guess these two definitions should somehow be equivalent (since Javanpeykar put them together in his notes).
So I wonder if the complement of the set of points of codim $1$ is of codim at least $2$? And if I would like to change $\textbf{Def 1}$ to actual integral points, do I just change the condition to all $s \in S$?
Sorry for multiple questions, any comment will help. Thanks!