How to query place names / POI in a given map tile or a long-lan?

Reading through dozens of samples and the API documentation, I still can’t figure out something which I thought would be dead simple.

How can I retrieve the place names that end-users see on a specific OSM tile, say in a given long/lat/radius?

E.g., in this location (40.713000, -74.010580):
I would like to return the place names / POI which are:
World Trade Center
World Trade Center Site
Ground Zero

That is - every place name that an end-user would see if they were to go to OSM and click on that location.
No need to return street names - only place names of points-of-interest such as the above.

I assume the API exposes this data somehow?

If someone could point to the right direction that would be great (something more specific than “check the API” would be appreciated…)