Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, and let $\pi:E\to X$ be the vector bundle defined by these data, so \begin{align} E:=\frac{\coprod_{i\in I}U_i\times\mathbb{R}^r}{\sim}, \end{align} where $\sim$ identifies $(p,\lambda)\in U_i\times\mathbb{R}^r$ and $(q,\mu)\in U_j\times\mathbb{R}^r$ iff $p=q$ and $\lambda=g_{ij}(p)(\mu)$.
I was confused by the following:
For any $p\in X$, the zero element $0_p\in E_p$ is the equivalence class of $(p,0)$, right? Since locally the bundle is trivial, it certainly holds that the zero is $(p,0)$ locally, and then on any overlap it holds thath $0_p=[p,e]=[p,0]$, $e\in\mathbb{R}^r$, so that $e=0$ since $g_{ij}(p)$ is invertible. Is this correct, or am I missing something?
The point is that $g_{ij}(p)$ is a linear map, thus $g_{ij}(p)(0)=0$, so if $p\in U_i\cap U_j$, you identify $(p,0)$ with $(p,g_{ij}(p)(0)=0)$. You can thus define the element $[p,0]$ which is the class of $(p,0)\in U_i\cap R^r, p\in U_i$ and this definition does not depend of $i$.