Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian: This is consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe I'm swapping left for right, I guess much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian.