In our class on stability theory we had the following defintion of Lyapunov stability:
Suppose we have a function $f:[a,\infty)\times G\to \mathbb{R}^n$ where $G$ is a connected open set in $\mathbb{R}^{n}$ and $a\in \mathbb{R}$. Moreover, suppose that for any $(t_0,x_0) \in [a,+\infty)\times G$, the initial value problem $$x'(t)=f\left(t,x(t)\right), x(t_0)=x_0$$ has unique solution, denoted by $\phi(t,t_0,x_0)$.
Then a solution $\psi(t)$ of $x'(t)=f\left(t,x(t)\right)$ is Lyapunov stable iff: $$\forall \epsilon>0, \forall t_0\ge a, \exists \delta>0, \forall x_0\in G: \|x_0-\psi(t_0)\|<\delta\implies (\forall t\ge t_0:\|\phi(t,t_0,x_0)-\psi(t)\|<\epsilon).$$
Now, I've also stumbled upon another definition of Lyapunov stability, which is:
Suppose we have a function $f:[a,\infty)\times G\to \mathbb{R}^n$ where $G$ is a connected open set in $\mathbb{R}^{n}$ and $a\in \mathbb{R}$. Moreover, suppose that for any $(t_0,x_0)\in[a,+\infty)\times G$, the initial value problem $$x'(t)=f\left(t,x(t)\right), x(t_0)=x_0$$ has unique solution, denoted by $\phi(t,t_0,x_0)$.
Then a solution $\psi(t)$ of $x'(t)=f\left(t,x(t)\right)$ is Lyapunov stable iff: $$\forall \epsilon>0, \exists \delta>0, \forall x_0\in G: \|x_0-\psi(a)\|<\delta\implies (\forall t\ge a:\|\phi(t,a,x_0)-\psi(t)\|<\epsilon)$$
My question is the following: Are these two definitions equivalent ? If not, which one of those is the standard one ?
How it was phrased the first definition is the standard definition of Lyapunov stability in the case of a non-autonomous system. It is true that in many books and courses one speaks about Lyapunov stability of an equilibrium (at least initially) of an autonomous system, and it allows to gloss over a lot of subtle details (since the equilibrium solution is, e.g., defined for all future times automatically), but the original definition is given for any solution existing on $[t_0,+\infty)$ (it can be trivially reduced to the stability of the trivial (pun intended) solution of a modified system).
The second definition is a confusing version of the first one (and strictly speaking is incorrect since it assumes that the initial time moment is always $t_0=a$). I would suggest not use the notes @KBS referenced in the comments.