Interesting fact: In the second part of the 1900s Kauri Gum was Auckland's main export. Its uses ranged hugely: chewing gum, jewellery resin, as a marine glue, as a fire starter and later as a varnish for fine violins!
Perhaps the most likely explanation is that the shoes were worn by the well to do folk for dancing, during the heyday of “taking the waters” at the various spa towns. The room where the spa water was dispensed, and in which social events such as dances were also held during the season, was commonly called the “pump room” – Hence “pumps” for the shoes which would often have been worn there.
Stilettos are named after the stiletto dagger. A long thin knife with a pointed needle like end.