Vert-de-gris would have had a special name because it was of commercial importance. Rust, less so.
In modern technical usage, the term refers to Copper(II) acetate monohydrate ("neutral verdigris") or Basic copper (II) acetate ("blue verdigris" or "green verdigris").
A 19th century (the 3rd French edition published in 1892) book (Traité de chimie industrielle by Johann Rudolf von Wagner) gives some uses that were in use at the time: pigment for oil paints and water color; to make Paris green, for dyeing cloth, and in gelding bronze (specifically, for giving a reddish finish to fire-gelded bronze).
Even at the time of writing, the author notes "verdigris is used much less often than previously, as it has been replaced by copper sulfate in many of its applications."
Going further back, the same substance is listed as a drug in Dioscorides' Materia Medica (1st century AD). The author notes that it's produced in Cyprus, which had mining industry in antiquity. The word "copper" itself comes from the place.