In the style of Gathmann's algebraic geometry notes, define a prevariety over an algebraically closed field $k$ as ringed space $(X,\mathcal{O}_X)$ such that 1) $X$ is irreducible, 2) $\mathcal{O}_X$ is a sheaf of $k$-valued functions, and 3) it has a finite open cover $\{U_i\}$, where each $(U_i, \iota^{-1}\mathcal{O}_X)$ is an affine variety ($\iota^{-1}\mathcal{O}_X$ is the pullback sheaf along the inclusion map $\iota: U_i \hookrightarrow X$). One then shows that irreducible, locally closed subsets of $X$ inherit a prevariety structure from $X$.
In my reading notes I wanted to go a little further and so I defined a subprevariety of $X$ to be any subset $Y \subseteq X$ such that $(Y, \iota^{-1}\mathcal{O}_X)$ is a prevariety. This seems like a natural definition of subobject here, and I believed that this definition captured all irreducible, locally closed subsets, i.e, there is a bijection between subprevarities and locally closed subsets of $X$. Then such sets would form the "embedded" prevarieties. However, I can't prove this fact and it's likely not true, since it seems most (all?) authors define subprevarieties to be locally closed. Of course, this could be analogous to how many authors define submanifolds (unqualified) as just the embedded subthings, so I didn't infer from this that there wasn't a bijection. My gut wants it to be true, however, since even a union of an open and closed subset can fail to be a subprevariety under the definition above.
Attempt 1: Let $Y \subseteq X$ be a subprevariety with open affine cover $\{U_i\}$. For each $U_i$ pick an open $V_i \subseteq X$ such that $U_i = V_i \cap Y$ and use the fact that $U_i$ is affine, and possibly that $V_i$ and $Y$ are irreducible, to prove that the intersection is closed in $V_i$. But I can't make any progress here since I'm lost on how to use the affine property.
Attempt 2: Show that $Y$ is open in it's closure by way of the finite affine cover, but I don't know how to proceed here either.
So, is this definition of subprevariety equivalent to being a locally closed subset of a prevariety? If not, what is an example of a subset satisfying the above definition that isn't locally closed?
The hope was that I could think of locally closed subsets playing the role of connected "embedded" prevarieties, analogous to submanifolds, and that relaxing the inverse image sheaf condition would capture the analog of immersed submanifolds.