In wikipedia a flow on a set $X$ is defined as
And then a flow associate with an autonomous systems of ordinary differential equations is defined as
As my understand, in the second definition, the set $S = \{\boldsymbol{x}(t)| t \in \mathbb{R}\} \subset \mathbb{R}^n$ play the role of $X$ in the first definition of a flow. And the flow associates with the autonomous systems of ordinary differential equations should be a mapping $\varphi :S \times \mathbb{R} \rightarrow S$. But in the second definition, just determine $\varphi $ at $S \times \boldsymbol{x}_0$. How to define $\varphi (\boldsymbol{x},t)$ with $\boldsymbol{x} \ne \boldsymbol{x}_0$? And how to prove: $\varphi (\varphi (\boldsymbol{x}_0,t),s) = \varphi (\boldsymbol{x}_0,s+t)$?
Edit: As @Arthur point out, $S = \mathbb{R}^n$.
If you want the flow to start from a different point, then you solve the differential equation with that other point as the initial point instead. Wherever you want your flow to start (i.e. whatever point you want to use as the first argument to $\varphi$), that's what you put as $\boldsymbol x_0$ when you solve the differential equation.
Note that this also makes all of $\Bbb R^n$ into the $X$ from the first definition, since any point of $\Bbb R^n$ is a valid initial point for the differential equation.