ca.analysis and odes - Are these two notions of Lipschitz hypersurface equivalent?

Let $S$ be a subset of $mathbb{R}^n$. I would like to call $S$

1. a Lipschitz(1) hypersurface if for every $xin S$ there is a hyperplane $H$ so that the orthogonal projection onto $H$ is a bi-Lipschitz map from a neighbourhood of $x$ in $S$ onto an open subset of $H$, and




2. a Lipschitz(2) hypersurface if for every $xin S$ there is a bi-Lipschitz map $psi$ from $Btimes(-1,1)$ onto a neighbourhood of $x$ in $mathbb{R}^n$ so that $psi^{-1}(S)=Btimes{0}$, where $B$ is an open subset of $mathbb{R}^{n-1}$.

It seems clear enough (*) that Lipschitz(1) implies Lipschitz(2). But is the converse true? And if not, what is a simple counterexample?

I have come across the notion of regions with Lipschitz boundaries in a number of papers on boundary value problems for PDEs. But every such paper seems to take the notion of Lipschitz-ness for granted.

(*) If $S$ is Lipschitz(1), then after a rotation of the axes, it locally looks like the graph of a Lipschitz function $gammacolonmathbb{R}^{n-1}tomathbb{R}$. Put $psi(x,t)=(x,gamma(x)-t)$ to obtain the Lipschitz(2) property.