Are heavy-bottom stock pots called something else?

Try searching for "laminated base pot", alternatively 'sandwich' or 'encapsulated'.

Other terms tend to be more trade markey, multiclad etc.

You could always buy a cheap pot & an even cheaper simmer ring instead ;)

Also known as marmite, yes, really, try searching for that.

Expect to spend a lot of money, maybe try an upmarket store. Those orange cast-iron ones from France are excellent.

I don’t know if the nomenclature is different in New Zealand, but you might consider an enameled ‘Dutch oven’. They’re typically cast iron, coated in enamel to make them easier to clean / less reactive. Unfortunately, they’re quite heavy, and they don’t tend to have the same proportions as a stock pot (they’re more squat versus the tall and skinny stock pots). And the ones from France tend to be quite expensive, but there are a lot of alternative brands these days

Another option is ‘tri-ply’ pans in which they look like stainless steel, but it’s actually stainless around a core of some more thermally conductive material, such as copper. ‘All clad’ is the original brand for this, but some more reasonable priced competitors have been available for the past 5 years or so

Then there are lots of brands that put a disk of copper or aluminum on the bottom of the pan. It’s usually pretty obvious on inspection— either a different color on the outside bottom, or a blatant disk that doesn’t smoothly transition to the sides of the pot

I would recommend going to a store, finding a sales person and specifically explaining what you want. It’s entirely possible that they won’t have any true ‘stock pots’ like you’re looking for, but they likely have pots in slightly different form factors that match your need. And you might want to add a spatter screen.