  We improve the model presented in Contini & Gu 2020 that describes the radial
mass distribution of brightest cluster galaxies (BCGs) and the diffuse
component also known as intra-cluster light (ICL), by assuming that the global
BCG+ICL radial mass distribution follows the sum of three profiles: a Jaffe and
an exponential profiles for the bulge and disk of the BCG, respectively, and a
modified version of an NFW profile for the ICL. We take advantage of a wide
sample of BCG+ICL systems simulated with our state-of-art semi-analytic model
to: (a) investigate the reliability of our BCG+ICL distribution by looking at
several scaling relations between the BCG+ICL stellar mass within different
apertures and the total BCG+ICL/halo mass, at different redshift; (b) make a
prediction of the distance where the radial distribution transitions from BCG
to ICL dominated. We find that our model nicely reproduces all the observed
scaling relations investigated at the present time with a compelling degree of
precision, but slightly biased-low with respect to observations at higher
redshifts ($z\gtrsim 0.5$). The transition radius predicted by our model is in
good agreement with recent observational results, and spans a range between
$\sim 15$ kpc and $\sim 100$ kpc. It mostly depends on the morphology of the
BCG, whether it is bulge or disk dominated, on the amount of ICL with respect
to the bulge and/or disk, and on the dynamical state of the group/cluster.
