In the context of Picard–Lindelöf theorem, one considers a function $f: D\rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}\times \mathbb{R}^n$, which is a function of time and space, $f = f(t, y)$. It is assumed that $f$ is continuous in $t$ and Lipschitz continuous in $y$.
It is unclear to me what exactly is meant by Lipschitz continuity in $y$. My intuition would say that it means that for every $t \in \mathbb{R}$, there exists a constant $L = L(t)>0$ such that $$ \left \vert f(t, y) - f(t, y') \right \vert \le L(t) \left \vert y-y' \right \vert$$ holds for every $y, y' \in \mathbb{R}^n$. That is, the functions $$y \mapsto f(t, y), \ t\in\mathbb{R},$$ are Lipschitz continuous.
However, many sources simply state that $L$ is a constant, which gives the impression that it does not have a dependency on $t$, i.e., Lipschitz in $y$ would mean: exists a constant $L > 0$ such that $$ \left \vert f(t, y) - f(t, y') \right \vert \le L \left \vert y-y' \right \vert$$ holds for every $y, y' \in \mathbb{R}^n$ and $t \in \mathbb{R}$.
What is the correct definition?