I encounter with two definitions for expansive continuous flows and their equivalence is unclear for me. Could anyone can explain for me please? Thanks in advance. I cite below these two definitions.

The first one in the paper "expansive one-parameter flow" of Bowen and Walters (1972): A flow $\varphi^t$ acting on a compact metric space $X$ is expansive if $\forall \epsilon, \exists \delta $ with the property that if $d(\varphi^t x, \varphi^{s(t)}y) \le \delta $ for all $t\in \mathbb{R}$ a pair of points $x,y \in X$ and a continuous function $s: \mathbb{R} \rightarrow \mathbb{R}$ with $s(0)=0$ then $y= \varphi^{t_0}x$ for some $|t_0| \le \epsilon.$

The second one in the book "Introduction to the modern theory of dynamical system" of Katok and Hasselblatt, page 125: A flow $\varphi^t$ acting on a compact metric space $X$ is expansive if there exists a positive $\delta$ such that any given continuous function $s: \mathbb{R} \rightarrow \mathbb{R}$ with $s(0)=0$ satisfying $d(\varphi^t x, \varphi^{s(t)}x) \le \delta $ for all $t\in \mathbb{R}$ then the property $d(\varphi^t x, \varphi^{s(t)}y) \le \delta $ implies $y= \varphi^{t_0}x$ for some $t_0 \in \mathbb{R}.$