Stadtteil/Bezirk & Stadt/City
What is the difference between these words? I have never heard a German city referred to with "city" it is always "Stadt".
Yes, a city is generally refered to as "Stadt" as in New York and "Stadtteil" means district (of a city) such as Brooklyn.
Colloquially, both "City" and "Stadt" are often used to refer to the city center, synonymously to the more specific word "Innenstadt".
A "Bezirk" or "Stadtbezirk" is an (administrative) district. Depending on what administrative district it refers to it can span less or more than one city districts or entire cities.
Hope this helps :)
Also refer to http://en.wikipedia.org/wiki/Stadtbezirk
Stadt means city. Stadttiel means a district. (or burrough). New York city would be a Stadt. Brooklyn would be a Stadttiel.
die Stadt=City der Stadtteil=part of a city (as teil means piece, like for cakes) der Bezirk=district, it depends on the dimension.
Bezirk is definitively no lake :)
You have many other words: das Umgebung, der Bereich... I think you'll learn practicing.