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, in this work, we revisit the microlensing effect that distorts the C IV broad emission line profile in J1004+4112. We take advantage of recent measurements of the image macro-magnification ratios, along with the fact that at one epoch, image B was not microlensed, thus constituting a reference spectrum to unambiguously characterize the microlensing effect observed in image A. After disentangling the microlensing in images A and B, we show that the microlensing-induced line profile distortions in image A, although variable, are remarkably similar over a period of 15 years. We find 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, characterized by either the full magnification profile of the C IV emission line or a set of four integrated indices, 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 C IV emission line in J1004+4112 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 C IV 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 C IV BLR half-light radius as r_1/2=2.8_−1.7^+2.0 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.