I cannot comment on your treaty article except to say that the US will consider it has the right to tax your income if you are a US citizen or green card holder. There is something in a treaty called the Savings Clause that gives the right of each country to tax its own residents as if no tax treaty existed. I don't know if there is totalization agreement with NZ so that you pay social security taxes in only one country. You might check that. For your foreign self-employment income as my TurboTax colleague Lina answered another taxpayer:
You would enter business deductions under the section"Schedule C". To report foreign earned income from self-employed, you would need to report in two places. One is under the "Schedule C" where you can report all your business income and deductions. The other place is under the "Foreign income". This will not double your income. Under the Foreign Income section, you will be prompted to answer questions related to the foreign earned income exclusion which reduces your taxable income and reflects on line 21 Schedule 1.
To enter under Schedule C:
Sign into your account and select Take me to my return
At the right upper corner, in the search box, type in schedule C and Enter
Select Jump to schedule C
On screen, Did you have any self-employment income or expenses? select Yes to continue
To enter under Foreign Income:
At the right upper corner, in the search box, type in foreign income and Enter
Select Jump to foreign income
