Two sets are called disjoint if they have no element in common. Thus for two subsets of $Y\subset X$ there is no distinction to be made between "disjoint in $Y$ and "disjoint in $X$".
Call two sets in a space separated if (they are disjoint and) neither one contains a limit point of the other. By the nature of the subspace topology, for two subsets of the subspace $Y\subset X$ there is no distinction between "separated in $Y$" and "separated in $X$". That is, a point of $Y$ in in the closure of a subset of $Y$ from the point of view of $X$ if and only if this is true from the point of view of $Y$. So there is no ambiguity in asking whether $Y$ can be expressed as the union of two nonempty separated sets.
But note that "having disjoint closures" is a stronger condition on two subsets of $Y$ Y$, for which there would be ambiguity.
(My parenthesis above was to ward off a real ambiguity of language: some people use "limit point of $A$" to mean any point in the closure of $A$; others do not include isolated points of $A$ as limit points.)
I wonder if the questioner was giving "disjoint" some topological meaning. I have noticed that topology students sometimes get the idea that it has such a meaning. This may be suggested by topologists' habit of using the expression "the disjoint union of spaces" for what is sometimes called the topological disjoint union or topological sum, i.e. the coproduct of objects in the category of spaces.