What is the connotation of “hamburger” in NZ?
This doesn’t directly answer your question, but in the U.K., hamburgers were often referred to as “burgers” or “beefburgers” rather than as “hamburgers” (except in the menus of American fast food chains). (I lived in the U.K. in 1987 through 1990, and I’ve visited five times since. It’s possible that this is no longer true.) Perhaps always referring to them as “hamburgers” was an indication that you were an American. Perhaps this is true in New Zealand.
I lived in NZ for several years in the early 1980s. “Hamburger” pretty much meant “hamburger,” although it frequently included what would be unusual toppings in the U.S., such as sliced beets (“beetroot”) or a fried egg (an “eggburger”). The texture and taste were also rather different from U.S. burgers.
I’ve also had sliced pineapple on hamburgers in NZ and Australia.
I’ve always been under the impression that hamburgers were so-named because the concept as we know it originated in Hamburg, Germany. If this was the case it would seem logical for them to be more or less universally referred to as hamburgers. This tale of their origin seems ubiquitous enough that one would think Cecil (or Snopes) would have already addressed it, however a search of the archives for “Hamburg Germany” only turns up a column about the origin of french fries , explaining that the French army officer credited for bringing them to the world was serving prison time in Hamburg when he came up with the idea. What a mysterious connection!
That’s not uncommon in the U.S., usually with teriyaki sauce.
Haj
…meat under the saddle… ewwhh!!!
…anyway, a hamburger in NZ, is a hamburger-I assume… the beetroot has fallen out of favour at a lot of fish’n’chip shops-but you can still find them around…
…egg burgers are very popular, while “tropical burgers” (with pineapple-at my cafe we called it the “Island Burger”) do pretty good business too…
the McDonalds New Zealand Burger is the Kiwi Burger, which contains both a fried egg and beetroot!
It used to be standard to have beetroot in some burgers, but then it stopped, then it came back, and now it’s fairly random. It can be a selling point, actually. Australia has this phenomenon too.
Burgers with a pineapple ring in it were called “Hawaiian Burgers”.
Whenever I see the word “Eggburger” I think of a TV comedy troupe from NZ called Funny Business who had these heavy metal fan characters. The punchline to the sketches almost always was “Aw, Clint! Ya Eggburger!”
Kiwis think of a Hamburger as a meat pattie in a bread bun, with salad and associated trimmings, sold from a fast-food place.
Minced (or ground) Beef is never called Hamburger - it’s called mince (d’uh). It took me ages to figure that one out.
Simon
Slight nit-pick … generally speaking (there may be exceptions) in the UK a beefburger is what we call the meat part of the hamburger *on its own * - in a bun we would call it a hamburger or just plain burger. 
And you mustn’t forget the Lamburger.
Mike Moore’s (former NZ Trade & Industry Minister, recent WTO Director-General) contribution to Kiwi culture.
This was before the Kebab arrived in NZ and took hold - by far the best way to mix lamb and salad and bread.
Most people thought this was a bit of a joke. I actually liked lamb mince for burgers - just a bit lighter than the traditional beef.
Nowadays, it’s just too pricey to mince up lamb for patties.
Simon