  J1004+4112 is a lensed quasar for which the first broad emission line profile
deformations due to microlensing were identified. Detailed interpretations of
these features have nevertheless remained controversial. Based on 15 spectra
obtained from 2003 to 2018, we revisit the microlensing effect that distorts
the CIV broad emission line profile. We show that the microlensing-induced line
profile distortions in image A, although variable, are remarkably similar over
a period of 15 years. They are characterized by a strong magnification of the
blue part of the line profile, a strong demagnification of the red part of the
line profile, and a small-to-negligible demagnification of the line core. We
used the microlensing effect to constrain the broad emission-line region (BLR)
size, geometry, and kinematics. For this purpose, we modeled the deformation of
the emission lines considering three simple, representative BLR models: a
Keplerian disk, an equatorial wind, and a biconical polar wind, with various
inclinations with respect to the line of sight. We find that the observed
magnification profile of the CIV emission line can be reproduced with the
simple BLR models we considered, without the need for more complex BLR
features. The magnification appears dominated by the position of the BLR with
respect to the caustic network -- and not by the velocity-dependent size of the
BLR. The favored models for the CIV BLR are either the Keplerian disk or the
equatorial wind, depending on the orientation of the BLR axis with respect to
the caustic network. We also find that the polar wind model can be discarded.
We measured the CIV BLR half-light radius as $r_{1/2} = 2.8^{+2.0}_{-1.7}$
light-days. This value is smaller than the BLR radius expected from the
radius-luminosity relation derived from reverberation mapping, but it is still
in reasonable agreement given the large uncertainties.
