Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t > t_0 = 0$$ Where I used the definition given here with $t_0 = 0$.
The classic example violating this (and back in the days sparking interest in chaotic dynamics) is the Lorenz System/Attractor. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs do not continuosly depend on the initial data.
Speaking of chaotic dynamical systems, you can also have chaotic behaviour in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.