Reading some articles and Silvermann's "The arithmetic of elliptic curves" I found these two different definitions for good ordinary reduction.
Silvermann, with Theorem V.3.1, says that an elliptic curve defined over a finite field with characteristic $p$ is ordinary if, for example, $E[p^r]=\mathbb{Z}/p^r\mathbb{Z}$ for every $r\in\mathbb{N}$. This implicitely says that an elliptic curve $E/K$ defined over a number field has (good) ordinary reduction at a place $v$ of $K$ if the reduction $\tilde{E}_v$ is ordinary.
From the other hand, I read in many articles that for elliptic curves $E/\mathbb{Q}$ the condition of having a (good) ordinary reduction at $p$ is equivalent to the condition \begin{equation*} p\nmid a_p:=p+1-|E(\mathbb{F}_p)|. \end{equation*}
Why is the last condition equivalent to the first?