If you read the NY rules carefully you will see that the NY definition of "limited use vehicle" (sometimes confusingly used interchangeably with "moped" regardless of the presence or absence of pedals) falls in three classes, A, B, and C, based on top speed as delivered by the manufacturer, not engine displacement.
The typical Vespa 50cc is NOT top-speed limited to 30mph so it will be flagged by the DMV based on its VIN as a limited use class A motorcycle. It requires motorcycle insurance, registration, and a motorcycle endorsement on your license. It does not require annual inspections.
If you buy a new Vespa 50cc, the dealer will tell you that it requires a motorcycle endorsement and they will handle the registration accordingly. If you buy a used one, you might be told anything or nothing that is relevant or true for NY state. This will catch up with you at the DMV, however, because of the VIN.
There exist some relatively new Vespa 50cc "LS" or "30mph" versions. The LS stands for "limited speed" and these do NOT require a motorcycle endorsement because it is a limited use motorcycle, Class B. I believe the limitation is established by the machine's computer and is difficult/impossible to override.
https://www.vespa.com/us_EN/models/primavera/primavera-30-mph-50-4s3v-2021/
The description specifically identifies NY as one of the states for which the model is intended because of the top speed limitation in the DMV rules, rather than the 50cc displacement that most other states use.
Many other 50cc machines (like the Genuine Buddy and Roughhouse 2-strokes) are delivered by the manufacturer with hardware restrictions that limit speed to 30mph to avoid the need for a motorcycle endorsement in states like NY. These restrictions are pretty easy to remove, and 2-strokes are fairly easy to soup up well beyond their stock configurations. As a result there are a whole lot of scooters out there that started life as speed-limited 50cc models ("Class B") but are no longer. They are clean in the DMV's eyes but are not really in conformance with the law. I don't think anybody particularly cares about this flagrant violation of regulation, but Piaggio/Vespa chose not to play this game.
Rules are different/vague for electric two-wheelers, but I don't think Revel rental scooters go faster than 30mph…although they seem to get to 30mph pretty quickly.