Induced Sheaf Structure is equivalent to Inverse Image Sheaf?

Question

Let $f:X \rightarrow Y$ be a map and let $Y$ be a ringed space, i.e. we have a sheaf of rings $O_Y$. Suppose that the regular functions are $k$-valued functions, where $k$ is a field. Define the open sets of $X$ to be generated by inverse images of open sets of $Y$. We want to give $X$ a ringed space structure. One way is to consider the inverse image sheaf. Another way is to define a function $g$ on $U$ to be regular, where $U$ is open in $X$, whenever there exists a covering $U \subseteq \cup_{a} f^{-1}(V_a)$ and regular functions of $Y$, $g_a : V_a \rightarrow k$, such that $g|_{U \cap f^{-1}(V_a)} = (g_a \circ f)|_{U \cap f^{-1}(V_a)}, \, \forall a$. Are the two constructions equivalent?

Answer 1

I ended up asking somebody in person, very known in algebraic geometry, and the two sheaf structures, i.e. the inverse image sheaf and the so called "induced structure" not only they are not equivalent, but very rarely coincide.