Difference between Lyapunov and uniform stability

The concept of uniform stability is mainly defined for non-autonomous systems, i.e. the systems of the form $$\dot x = f(t,x),$$ but Lyapunov stability or what is often called just stability is defined for both autonomous systems and non-autonomous systems.

We first assume that the system is non-autonomous and let $x_0(t)$ be the initial condition of $\dot x = f(t,x)$. Then we have the following definitions [Khalil, Hassan K., Nonlinear systems., Upper Saddle River, NJ: Prentice Hall. xv, 750 p. (2002). ZBL1003.34002.]:

• $\textbf{Lyapunov stability (stability)}$: for each $\varepsilon>0$, there is $\delta=\delta(\mathbf{\epsilon,t_0})>0$ such that $$\|x(t_0)-x_0(t_0)\|<\delta~~\Rightarrow ~~\|x(t)-x_0(t)\|<\epsilon, ~~\forall t\geq t_0\geq0$$
• $\textbf{Uniform stability}$: for each $\varepsilon>0$, there is $\delta=\delta(\mathbf{\epsilon})>0$, $\textbf{independent of}$ $\mathbf{t_0}$, such that $$\|x(t_0)-x_0(t_0)\|<\delta~~\Rightarrow ~~\|x(t)-x_0(t)\|<\epsilon, ~~\forall t\geq t_0\geq0$$

Note that the only difference is in the definition of $\delta$, where for uniform stability, $\delta$ is independent of $t_0$.

For autonomous system, $\dot x = f(x)$, uniform stability is the same as Lyapunov stability because $f$ is not explicitly a function of $t$.