  Luminosity determination within the ALICE experiment is based on the
measurement, in van der Meer scans, of the cross sections for visible processes
involving one or more detectors (visible cross sections). In 2015 and 2018, the
Large Hadron Collider provided Pb$-$Pb collisions at a centre-of-mass energy
per nucleon pair of $\sqrt{s_{\rm NN}} = 5.02$ TeV. Two visible cross sections,
associated with particle detection in the Zero Degree Calorimeter (ZDC) and in
the V0 detector, were measured in a van der Meer scan. This article describes
the experimental set-up and the analysis procedure, and presents the
measurement results. The analysis involves a comprehensive study of
beam-related effects and an improved fitting procedure, compared to previous
ALICE studies, for the extraction of the visible cross section. The resulting
uncertainty of both the ZDC-based and the V0-based luminosity measurement for
the full sample is 2.5%. The inelastic cross section for hadronic interactions
in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV, obtained by efficiency
correction of the V0-based visible cross section, was measured to be $7.67 \pm
0.25$ b, in agreement with predictions using the Glauber model.
