Yeah the other day I was looking up Long Island cos i was watching a documentary set there and I wanted to get a bit of a handle on the geography and stuff, and I got the impression that, similarly to what you're saying here, when people say "Long Island" they are not really meaning the whole island but tend to be referring to parts of it which don't fall under other names and haven't been claimed by other places. Is that correct?
Yeah, Long Island is an area, not a town in itself. It's made up of cities, towns and villages.
when you cross the east river from Manhattan, you land in two NYC boroughs: Queens and Brooklyn. as you continue east, you leave those boroughs and hit towns in Nassua and the Suffolk counties in New York State, which make up Long Island. The furthest eastern portions of Long Island is the area known as the Hamptons, made up of towns where rich NYC folks have summer houses.