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(Abridged) We study the galaxy mass-size relation in CARLA spectroscopically confirmed clusters at $1.4<z<2.8$, which span a total stellar mass $11.3<\mathrm{log}(M^c_*/M_{\odot})<12.6$ (halo mass $13.5 \lesssim \mathrm{log}(M^c_h/M_{\odot}) \lesssim 14.5$). Our main finding is that cluster passive ETG at $z \gtrsim 1.5$ with ${\rm log}(M/M_{\odot})>10.5$ are systematically $\gtrsim 0.2-0.3~{\rm dex}$ larger than field ETGs. The passive ETG average size evolution is slower at $1<z<2$ when compared to the field. This could be explained by differences in the formation and early evolution of galaxies in haloes of a different mass. Strong compaction and gas dissipation in field galaxies, followed by a sequence of mergers may have also played a significant role in the field ETG evolution, but not in the evolution of cluster galaxies. Our passive ETG mass-size relation shows a tendency to flatten at $9.6<{\rm log}(M/M_{\odot})<10.5$, where the average size is $\mathrm{log}(R_e/\mathrm{kpc}) = 0.05 \pm 0.22$. This implies that galaxies in the low end of the mass-size relation do not evolve much from $z\sim 2$ to the present, and that their sizes evolve in a similar way in clusters and in the field. BCGs lie on the same mass-size relation as satellites, suggesting that their size evolution is not different at redshift z $\gtrsim$ 2. Half of the active ETGs ($\sim 30\%$ of the ETGs) follow the field passive galaxy mass-size relation, and the other half follow the field active galaxy mass-size relation. These galaxies likely went through a recent merger or neighbor galaxy interaction, and would most probably quench at a later epoch and increase the fraction of passive ETGs in clusters. We do not observe a large population of compact galaxies, as is observed in the field at these redshifts, implying that the galaxies in our clusters are not observed in an epoch close to their compaction.