Smooth sections of smooth vector bundle

Question

Suppose that $E \to B$ is a (real for example) smooth vector bundle ($B$ is assumed to be a smooth manifold). There is a important notion of the smooth section $s:B \to E$: is has to satisfy $s(x) \in E_x$ where $E_x=p^{-1}(x)$ is the fiber over $x$. Each vector bundle is equipped with so called trivializations and one could understand the smoothness of $s$ in the following way: when $x \in B$ there is a small neighborhood $U$ of $x$ and a local trivialization $\varphi:p^{-1}(U) \to U \times \mathbb{R}^n$ such that $\varphi \circ s$ is smooth as a map from $B$ to $U \times \mathbb{R}^n$ (this does make sense). But as far as I know one can view $E$ as a smooth manifold by its own: therefore one could define a smooth section $s:B \to E$ simply as a smooth map between manifolds, such that $s(x) \in E_x$ for each $x \in B$. Are these two notions equivalent?