"nt" pronounced as "n" in American English (as in "Internet"): what is it called?

The Wikipedia article on intervocalic alveolar flapping addresses this directly:

The cluster [nt] can also be flapped/tapped; the IPA symbol for a nasal tap is [ɾ̃]. As a result, in quick speech, words like winner and winter can become homophonous. Flapping/tapping does not occur for most speakers in words like carpenter and ninety, which instead surface with [d].

The name for the phenomenon would be “intervocalic alveolar flapping of [nt] clusters”

If what you are hearing is [ˌɪnɚˈnæʃənɫ̩], where /nt/ becomes [n], then that one is called assimilation, and it is by no means restricted to the United States alone. Specifically, this is a case of progressive assimilation (left-to-right), in which a later sound becomes more like an earlier one.

It is possible that you should also take into account that a /t/ may often be expressed as a glottal [ʔ] in the syllable coda, and so may appear to you to have been altogether deleted here — especially if the stop is incomplete, as in rapid speech it may well be.

In fact, it is not uncommon to hear “international” pronounced as [ˌɪ̃ʔɚˈnæʃənɫ̩]. Here the /n/ nasalizes the preceding vowel in regressive assimilation (right-to-left), but is itself deleted.

This might even be a case of reciprocal assimilation where the bleed-over goes both ways.

The T isn't being pronounced as an N in those words. Rather, it's being elided entirely, and the neighboring N-sound is what we hear in its place.

To my ear, the T-sound isn't being transformed to match neighboring sounds, in the way of an assimilation. Additionally, the loss of the T-sound occurs even when speaking slowly, in contrast to the explanation of how the intervocalic alveolar flapping phenomenon occurs.

There are other instances in English where the T is transformed into a glottal stop or nasally released, in a process called debuccalization.

Examples of this from the above Wikipedia article are "get ready" and "cotton." If you compare the pronunciations of these words to "internet" with an elided T, you should be able to hear that these are different phenomena.