Context. Galaxies in dense environments are subject to interactions and mechanisms that directly affect their evolution by lowering their gas fractions and consequently reducing their star-forming capacity earlier than their isolated counterparts. Aims. The aim of our project is to get new insights into the role of environment in the stellar and baryonic content of galaxies using a kinematic approach, through the study of the Tully-Fisher relation (TFR).Methods. We study a sample of galaxies in eight groups, over-dense by a factor larger than 25 with respect to the average projected density, spanning a redshift range of 0.5 < z < 0.8 and located in ten pointings of the MAGIC MUSE Guaranteed Time Observations program. We perform a morpho-kinematics analysis of this sample and set up a selection based on galaxy size, [O II]λλ3727,3729 emission line doublet signal-to-noise ratio, bulge-to-disk ratio, and nuclear activity to construct a robust kinematic sample of 67 star-forming galaxies. Results. We show that this selection considerably reduces the number of outliers in the TFR, which are predominantly dispersion-dominated galaxies. Similar to other studies, we find that including the velocity dispersion in the velocity budget mainly affects galaxies with low rotation velocities, reduces the scatter in the relation, increases its slope, and decreases its zero-point. Including gas masses is more significant for low-mass galaxies due to a larger gas fraction, and thus decreases the slope and increases the zero-point of the relation. Our results suggest a significant offset of the TFR zero-point between galaxies in low- and high-density environments, regardless of the kinematics estimator used. This can be interpreted as a decrease in either stellar mass by ∼0.05 − 0.3 dex or an increase in rotation velocity by ∼0.02 − 0.06 dex for galaxies in groups, depending on the samples used for comparison. We also studied the stellar and baryon mass fractions within stellar disks and found they both increase with stellar mass, the trend being more pronounced for the stellar component alone. These fractions do not exceed 50%. We show that this evolution of the TFR is consistent either with a decrease in star formation or with a contraction of the mass distribution due to the environment. These two effects probably act together, with their relative contribution depending on the mass regime.