Among carbon-enhanced metal-poor (CEMP) stars, some are found to be enriched in slow-neutron capture (s-process) elements (and are then tagged CEMP-s), some have overabundances in rapid-neutron capture (r-process) elements (tagged CEMP-r), and some are characterized by both s- and r-process enrichments (tagged CEMP-rs). The current distinction between CEMP-s and CEMP-rs is based on their [Ba/Fe] and [Eu/Fe] ratios, since barium and europium are predominantly produced by the s- and the r-process, respectively. 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 propose a more robust classification method for CEMP-s and CEMP-rs stars using additional heavy elements other than Ba and Eu. Once a secure classification is available, it should then be possible to assess whether the i-process or a variant of the s-process better fits the peculiar abundance patterns of CEMP-rs stars. We analyse high-resolution spectra of 24 CEMP stars and one r-process enriched star without carbon-enrichment, observed mainly with the high-resolution HERMES spectrograph mounted on the Mercator telescope (La Palma) and also with the UVES spectrograph on VLT (ESO Chile) and HIRES spectrograph on KECK (Hawaii). Stellar parameters and abundances are derived using MARCS model atmospheres. Elemental abundances are computed through spectral synthesis using the TURBOSPECTRUM radiative transfer code. Stars are re-classified as CEMP-s or -rs according to a new classification scheme using eight heavy element abundances. Within our sample of 25 objects, the literature classification is globally confirmed, except for HE 1429-0551 and HE 2144-1832, previously classified as CEMP-rs and now as CEMP-s stars. The abundance profiles of CEMP-s and CEMP-rs stars are compared in detail, and no clear separation is found between the two groups; it seems instead that there is an abundance continuum between the two stellar classes. There is an even larger binarity rate among CEMP-rs stars than among CEMP-s stars, indicating that CEMP-rs stars are extrinsic stars as well. The second peak s-process elements (Ba, La, Ce) are slightly enhanced in CEMP-rs stars with respect to first-peak s-process elements (Sr, Y, Zr), when compared to CEMP-s stars. Models of radiative s-process nucleosynthesis during the interpulse phases reproduce well the abundance profiles of CEMP-s stars, whereas those of CEMP-rs stars are explained well by low-metallicity 1M_{sun}_ models experiencing proton ingestion. 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. 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 Hertzsprung-Russell (HR) diagram using Gaia DR2 parallaxes; they are found to lie mostly on the red giant branch (RGB). CEMP-rs stars present most of the characteristics of extrinsic stars such as CEMP-s, CH, barium, and extrinsic S stars; they can be explained as being polluted by a low-mass, low-metallicity thermally-pulsing asymptotic giant branch (TP-AGB) companion experiencing i-process nucleosynthesis after proton ingestion during its first convective thermal pulses. As such, they could be renamed 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.