Did you have to renounce your New Zealand citizenship? I think a lot of Americans are uncomfortable with the idea of a person owing allegiance to more than one country.
No, in fact I still travel on my NZ passport as I haven’t got around to getting an Australian one.
I hope they get over it. It must be extremely common for naturalized Americans to hold onto their previous citizenship. I for one have both a South African and American passport.
I also have dual Australian/NZ citizenship and didn’t have to renounce my NZ citizenship when I became Australian, which is just as well frankly.
It’s not really a big deal (both countries recognise the Queen as head of state, after all) and heaps of people I know in both countries hold dual citizenships, not only Aust/NZ but also Aust/UK or NZ/UK, Aust/Canada, NZ/Canada, etc. One guy I knew years ago had the trifecta of triple citizenship - UK, New Zealand and Australia.
Interesting. What are American tourists called, then - those who are just visiting and have no intention of settling in Israel?
“Annoyances”. 
Heh. Generally, they’re just called “American Tourists”
On a related note, Jewish immigrants from anglophone countries (the U.S., Canada, UK, Australia and South Africa) are often lumped together under “Anglosaxim”,a name many of us find hilarious.