The Italy/New Zealand treaty says:
...shall be exempt from tax in the first mentioned Contracting State on any remuneration for such teaching, advanced study or research in respect of which he is subject to tax in the other Contracting State.
While the Italy/US treaty says:
shall, for a period not exceeding two years, be exempt from tax in the first-mentioned Contracting State in respect of remuneration from such teaching or research.
The difference is that in the NZ treaty the exemption is explicitly conditioned on the teacher being taxed in the other country, but in the US treaty it is not so conditioned. But the US treaty does not grant an exemption in the "second" state (country), that is the country of residence of the teacher. It looks to me that in any normal case the teacher would owe the normal tax to his or her resident country (ital in this case). If for some other reason part of this income was exempt from tax in the teacher's home country, NZ would them tax it, while apparently the US would not. But it is hard to see what other reason would be likely to apply.
I do not see anything in these treaty provisions which would say that the teacher should not pay the normal tax in the teacher's home county.