Two conditions are being confused here, because the second one is less known and is inconsistently named.
- A function $f$ (either scalar or vector valued) is Lipschitz if there is a constant $L$ such that $\|f(x)-f(y)\|\le L\|x-y\|$ for all $x,y$. This is denoted by $f\in\mathrm{Lip}(U)$ or $f\in C^{0,1}(U)$ where $U$ is the domain of $f$.
- A function $f$ is Lipschitz smooth or Lipschitz continuously differentiable if there is a constant $L$ such that $\|\nabla f(x)-\nabla f(y)\|\le L\|x-y\|$ for all $x,y$. This is denoted by $\nabla f\in\mathrm{Lip}(U)$ or $f\in C^{1,1}(U)$ where $U$ is the domain of $f$.
That is, the second condition is the Lipschitz continuity of first-order derivatives. A sufficient condition for $f\in C^{1,1}$ is that the second derivative of $f$ is bounded: see $C^{1,1}$ regularity.