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Among Carbon-Enhanced Metal-Poor (CEMP) stars, some are found to be enriched in s-process elements (CEMP-s), in r-process elements (CEMP-r) or in both s- and r-process elements (CEMP-rs). The origin of the abundance differences between CEMP-s and CEMP-rs stars is presently unknown. It has been claimed that the i-process, whose site still remains to be identified, could better reproduce CEMP-rs abundances than the s-process. We analyze high-resolution spectra of 25 metal-poor stars, observed with the high-resolution HERMES spectrograph mounted on the Mercator telescope, La Palma, or with the UVES/VLT and HIRES/KECK spectrographs. We propose a new, robust classification method for CEMP-s and CEMP-rs stars using eight heavy element abundances. The abundance profiles of CEMP-s and CEMP-rs stars are derived and there appears to be an abundance continuum between the two stellar classes. CEMP-rs stars present most of the characteristics of extrinsic stars such as CEMP-s, CH, Barium and extrinsic S stars, with an even larger binarity rate among CEMP-rs stars than among CEMP-s stars. Stellar evolutionary tracks of an enhanced carbon composition (consistent with our abundance determinations) are necessary to explain the position of CEMP-s and CEMP-rs stars in the HR diagram using Gaia DR2 parallaxes; they are found to lie mostly on the RGB. CEMP-rs stars can be explained as being polluted by a low-mass, low-metallicity TP-AGB companion experiencing i-process nucleosynthesis after proton ingestion during its first convective thermal pulses. The global fitting of our i-process models to CEMP-rs stars is as good as the one of our s-process models to CEMP-s stars. As such, CEMP-rs stars could be renamed as CEMP-sr stars, since they represent a particular manifestation of the s-process at low-metallicities. For these objects a call for an exotic i-process site may not necessarily be required anymore.