  The peeling process, which describes a step-by-step exploration of a planar
map, has been instrumental in addressing percolation problems on random
infinite planar maps. Bond and face percolation on maps with faces of arbitrary
degree are conveniently studied via so-called lazy-peeling explorations. During
such explorations distinct vertices on the exploration contour may at later
stage be identified, making the process less suited to the study of site
percolation. To tackle this situation and to explicitly identify
site-percolation thresholds, we come back to the alternative "simple" peeling
exploration of Angel and uncover deep relations with the lazy-peeling process.
Along the way we define and study the random Boltzmann map of the half-plane
with a simple boundary for an arbitrary critical weight sequence. Its
construction is nontrivial especially in the "dense regime" where the
half-planar random Boltzmann map does not possess an infinite simple core.
