Confused about the definition of open subscheme in Hartshorne

Question

Let $X$ be a scheme and $U$ an open subset. In Exercise 2.2 in Chapter II, Hartshorne states "we refer to $(U, O_X|_U)$ as an open subscheme of X", but in the definition on p.85 he defines open subschemes as an open subset $U$ of $X$ with structure sheaf $O_U$ isomorphic to $O_X|_U$. I am not understanding the reason for this. What is the problem with defining open subschemes as $(U, O_X|U)$ as in (II Ex 2.2)?