The definition given in the link is sometimes called "weakly Lipschitz".
The two definitions given by Pietro Majer are indeed equivalent, they are sometimes called "strongly Lipschitz" and "strong cone condition", respectively.

If the boundary of the domain is compact, then any cover of the boundary has a finite subcover. So in all the definitions, it doesn't matter whether we state the property for a neighborhood of every point, or for a finite cover.

One can easily see that strongly Lipschitz implies weakly Lipschitz.
However, the converse is not true. This has to do with the fact that the implicit function theorem does not hold for Lipschitz continuous functions.
E.g., there exist polyhedra that are are weakly Lipschitz but not strongly Lipschitz; they fail to be a *graph* of a Lipschitz continious function.
For more see P. Grisvard, Elliptic Problems on Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985.

Unfortunately, in the literature "Lipschitz" is occasionally used meaning "strongly Lipschitz", but sometimes meaning "weakly Lipschitz".