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Résumé : We present the largest uniform study to date of 50 thermonuclear Type Ia supernovae interacting with circumstellar medium (SNe Ia-CSM). As the first paper in a series delineating the population properties of this class, this work focuses specifically on a uniform analysis of their light curves. Our results show that SNe Ia-CSM display a significantly more uniform light-curve morphology than interacting core-collapse SNe IIn, closely resembling the luminous-slow subgroup. The Ia-CSM sample spans approximately an order of magnitude in peak luminosity, total timescale, and radiated energy, with a median optical peak luminosity of $\approx2\times10^{43}$ erg s$^{-1}$ ($\approx-20$ mag). We find a distinct bimodality in the radiated energy distribution: a predominant population of energetic events with $\approx 2\times10^{50}$ erg and characteristic timescales of $\approx100$ days, and a secondary population that is roughly an order of magnitude less energetic with shorter timescales of $\approx50$ days. Decomposition of the luminosity contributions from underlying SN Ia and CSM interaction by the Ia-CSM spectral classification reveals a diverse range of underlying SN Ia subtypes spanning from luminous 1991T-like to faint 1991bg-like. Both SN Ia and CSM properties are broadly consistent with the core-degenerate channel resulting from common-envelope evolution followed by core merger, where the presence of dense CSM is dictated by the delay time between the common-envelope phase and the thermonuclear explosion. Subsequent papers in this series will utilize numerical simulations, semi-analytical modeling, spectroscopy, and host-galaxy environments to further constrain the progenitor and explosion pathways to SNe Ia-CSM.