In answer to the original question, I got this from the Long Island Staylace Association! (I am bemused by the amount of corset websites to be found).

Stays: (a pair of) C17th and c18th term for the boned underbodice previously known as a "pair of bodies." The term persisted into the c19th but was more usually replaced by its French equivalent, the "corset." The term was also applied to the stiff inserts of whalebone or steel which shaped this garment.

The OED has: a corset made of two pieces laced together and stiffened by strips of whalebone.

That's certainly how we used the word in our family.

Some of the sites I've looked at have said one of the functions of these things were to make the lady walk upright and gracefully, so maybe that's why they give her "balance". Or maybe they've restrained her bad behaviour? I thought "ballast" made sense as her waist would have been thickened because she'd had a baby.