We present observations of SN 2021csp, the second example of a newly-identified type of supernova (Type Icn) hallmarked by strong, narrow, P Cygni carbon features at early times. The SN appears as a fast and luminous blue transient at early times, reaching a peak absolute magnitude of -20 within 3 days due to strong interaction between fast SN ejecta (v ~ 30000 km/s) and a massive, dense, fast-moving C/O wind shed by the WC-like progenitor months before explosion. The narrow line features disappear from the spectrum 10-20 days after explosion and are replaced by a blue continuum dominated by broad Fe features, reminiscent of Type Ibn and IIn supernovae and indicative of weaker interaction with more extended H/He-poor material. The transient then abruptly fades ~60 days post-explosion when interaction ceases. Deep limits at later phases suggest minimal heavy-element nucleosynthesis, a low ejecta mass, or both, and imply an origin distinct from that of classical Type Ic supernovae. We place SN 2021csp in context with other fast-evolving interacting transients, and discuss various progenitor scenarios: an ultrastripped progenitor star, a pulsational pair-instability eruption, or a jet-driven fallback supernova from a Wolf-Rayet star. The fallback scenario would naturally explain the similarity between these events and radio-loud fast transients, and suggests a picture in which most stars massive enough to undergo a WR phase collapse directly to black holes at the end of their lives.