Here's one definition, from A Short Course in Ordinary Differential Equations, Qingkai Kong (2014):
Here's another definition from Ordinary Differential Equations, Jack K. Hale (1980):
Denote the first definition by $(1)$, and the second by $(2)$. It is obvious that $(2)\implies (1)$, because if the Lipschitz condition is satisfied on any closed bounded subset of $D$, then it must be satified on a closed neighborhood of any given point in $D$ (does it require some extra assumptions about $D$ for any point to have a closed neighborhood?). But it isn't obvious to me why $(1)\implies (2)$. How to prove that these two definitions are equivalent?
EDIT: It seems that I can't have embedded images in my post yet, so here's the two definitions mentioned above (I changed the notation to make it more consistent):
Let $f:D\subset \mathbb R\times\mathbb R^n\to\mathbb R^n$. $f$ is locally Lipschitz in $x$ on $D$, if
Definition (1) For any $(t^*,x^*)\in D$, there is a neighborhood $\mathcal N^*$ of $(t^*,x^*)$ and a constant $k>0$ such that $$|f(t,x_1)-f(t,x_2)|\leq k\lvert x_1-x_2\rvert,\ \forall(t,x_1),(t,x_2)\in\mathcal N^*\cap D.$$
Definition (2) For any closed bounded set $U\subseteq D$ there is a $k>0$ such that $$|f(t,x_1)-f(t,x_2)|\leq k|x_1-x_2|,\ \forall(t,x_1),(t,x_2)\in U.$$