  We classify several notions of norm attaining Lipschitz maps which were
introduced previously, and present the relations among them in order to verify
proper inclusions. We also analyze some results for the sets of Lipschitz maps
satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For
instance, we characterize a Banach space $Y$ with the Radon-Nikod\'ym property
in terms of the denseness of norm attaining Lipschitz maps with values in $Y$.
Further, we introduce a property called the local directional
Bishop-Phelps-Bollob\'as property for Lipschitz compact maps, which extends the
one studied previously for scalar-valued functions, and provide some new
positive results.
