score:39
There are a few things called Luxembourg and that might create some confusion:
Now, Schengen is a village on the Luxembourgish side of the border tri-point between Luxembourg, France, and Germany. The agreement itself was signed on a riverboat, so as to symbolically be “on the border” without knowing exactly in which country (although for legal purpose, it's considered to be Schengen and Luxembourg is the depositary of the original treaty). The boat, called “Princesse Marie-Astrid”, still does cruises on the Moselle river.
Finally, Schengen is very small, I don't think the area around it was ever known as “Schengen area” in any of the local languages. What's now called “Schengen area” is typically the whole 20+ country strong border-free area.
Upvote:4
"Luxembourg" refers to various parts of a former medieval Duchy of that name.
The namesake part of the Duchy is a country of that name. To the west is a former part of the country that was occupied by Belgium in 1839, and is the (Belgian) province of Luxembourg.
"Luxembourg" could refer to one (of three) districts in the country that goes by that name.
Finally, it could refer to the capital city (located in the district mentioned above).