I am currently reading the book "Controlled Markov Processes and Viscosity solutions" by Fleming and Soner. In the first chapter about deterministic controls, they introduce a differentiation between stopping times from a closed and an open set. They consider the cost functional \begin{align} J(t_0,x_0;u) = \int_{t_0}^{\tau} L(s,x(s), u(s)) ds + g(\tau,x(\tau)) \chi_{\tau < t_1}+ \psi(x(t_1)) \chi_{\tau \geq t_1} \label{OptB} \end{align} the function $g$ is assumed to be continuous, and standard control equation\begin{align*} \begin{cases} &{x}_t(t)=f(t,x(t),u(t)),\\ &x(t_0)=x_0 \end{cases} \end{align*} In order to guarantee the existence of a unique solution, they assume \begin{align*} ||f(t,x,v)-f(t,y,v)|| < L ||x-y|| \quad \text{ for all }t\in[t_0,t_1],x,y \in \mathbb{R}^n. \end{align*}
Concretely, they make a distinction between considering the exit from an open set $O \subset \mathbb{R}^n$ or from its closure $\overline{O}$. Consider first the exit time from $\overline{O}$ \begin{align} \tau := \begin{cases} \inf \{s \in [t_0,t_1) : x(s) \not \in \overline{O} \} \\ t_1, \text{ if } x(s) \in \overline{O} \text{ for all } s\in [t_0,t_1] \end{cases} \label{closed_stop_time} \end{align} or alternatively the exit time from the open set $O$ \begin{align} \tau := \begin{cases} \inf \{s \in [t_0,t_1) : x(s) \in \partial O \} \\ t_1, \text{ if } x(s) \in O \text{ for all } s\in [t_0,t_1). \end{cases} \label{open_stop_time} \end{align} However, both definitions imply $(\tau , x(\tau)) \in ([t_0,t_1)\times \partial O)\cup (\{t_1 \}\times \overline{O})$ as long as $x$ is assumed to be continuous. \ I am aware, that the \textit{cost} associated with $x(.)\in \partial O$ depends on the choice of the stopping times. In the case of equation (\eqref{closed_stop_time}) we associate $x(.)\in \partial O$ with the running cost $L$, while in the case of equation (\eqref{open_stop_time}) we associate it with the boundary cost $g$. Although this is useful for understanding the modeling, it remains unclear why such a distinction would be necessary. \
The distinction also does not seem to have any effect on the proof techniques besides some stricter regularity assumptions regarding the HJB-equations and verification theorems. Why would they make such a distinction? What am I missing?
EDIT: Note that what I referred to as stop or stopping time seems to be called exit time in the area of toptimal control.