I am a little confused by the term family of hypersurfaces. For simplicity, let $Y$ be a projective curve (and also given the embedding, so we can talk about the line bundle $\mathcal O_Y(1)$). When I say $X \to Y$ is a (flat) family of hypersurfaces in $\mathbb P^n$ of degree $d$, I may mean the following:
(i) $\pi\colon X \to Y$ is a (flat) morphism with every fiber isomorphic to some hypersurface in $\mathbb P^n$ of degree $d$. (I believe the flatness is followed from the fact that each fibre is isomorphic to some hypersurface in $\mathbb P^n$ of degree $d$, at least under some very weak assumption)
(ii) The conditions in (i), moreover we assume $X\subset Y\times \mathbb P^n$ and $\pi$ is the restriction of the first projection.
(iii) The conditions in (ii), moreover we assume $X\subset Y\times \mathbb P^n$ and $\pi$ is given by zero locus of some secion in $\mathcal O_Y(1)\boxtimes\mathcal O_{\mathbb P^n}(d)$.
(iv) The conditions in (iii), moreover we assume the embedding of $Y$ is $Y\subset \mathbb |\mathcal O_{\mathbb P^n}(d)|$.
I want to know that, are those contions all equivalent?