We report comprehensive multi-wavelength observations of a peculiar Type Ia-like supernova ("SN Ia-pec") ASASSN-15pz. ASASSN-15pz is a spectroscopic "twin" of SN 2009dc, a so-called "Super-Chandrasekhar-mass" SN, throughout its evolution, but it has a peak luminosity M_B,peak = -19.69 +/- 0.12 mag that is \approx 0.6 mag dimmer and comparable to the SN 1991T sub-class of SNe Ia at the luminous end of the normal width-luminosity relation. The synthesized Ni56 mass of M_Ni56 = 1.13 +/- 0.14 M_sun is also substantially less than that found for several 2009dc-like SNe. Previous well-studied 2009dc-like SNe have generally suffered from large and uncertain amounts of host-galaxy extinction, which is negligible for ASASSN-15pz. Based on the color of ASASSN-15pz, we estimate a host extinction for SN 2009dc of E(B-V)_host=0.12 mag and confirm its high luminosity (M_B, peak2009dc \approx -20.3 mag). The 2009dc-like SN population, which represents ~1% of SNe Ia, exhibits a range of peak luminosities, and do not fit onto the tight width-luminosity relation. Their optical light curves also show significant diversity of late-time (>~ 50 days) decline rates. The nebular-phase spectra provide powerful diagnostics to identify the 2009dc-like events as a distinct class of SNe Ia. We suggest referring to these sources using the phenomenology-based "2009dc-like SN Ia-pec" instead of "Super-Chandrasekhar SN Ia," which is based on an uncertain theoretical interpretation.