> They should either call them "preorders", like everyone else does, or make up some brand new term if they think "preorder" is too ugly.

Some people call them _quasi orders_. In particular, Isabelle/HOL has a quasi_order axiom class in its [Orderings theory](https://isabelle.in.tum.de/dist/library/HOL/HOL/Orderings.html). Some people working in combinatorics also use this term ([here](https://en.wikipedia.org/wiki/Well-quasi-ordering#Notes) is a short bibliography).

Preorder is still far more common. Unlike Isabelle/HOL, Coq uses the conventional term (see [Coq.Classes.RelationClasses](https://coq.inria.fr/library/Coq.Classes.RelationClasses.html)).