Well, the original form was strictly for countries; if you want to use it to describe something that is neither on the level of analysis nor on the axis of variation of the original form, its not worth even using the terminology (the common modern form still uses the country unit of analysis, but is an economic rather than geopolitical axis.)
> Take Brazil for example; the difference between middle class suburbs in Sao Paulo and a remote settlement in an Amazonian state straddle our concepts of First and Third World.
Not really. Actually, have sharp geographic and class divides is typical of the developing world; having elite areas is not at all out of line of the traditional understanding of a developing (or "Third World") country.
Though a binary developing/developed or First/Third divide is problematic for other reasons -- Brazil (like Argentina, Mexico, and lots of other places) really ought to be viewed in a middle tier; its nearly as far above, by most meaures, a lot of the places more typical of the "Third World" as it is behind the places more typical of the "First World". (Perhaps we ought to resurrect "Second World" for this.)
> For many people in American cities, their access to healthcare, education, sanitation is not different, or in fact worse, than it would be in cities more conventionally thought of as third world.
To the extent one wants to discuss claims that that is the case, it may make sense to describe the particular American cities being like developing (or "Third") world cities with regard to the particular concerns, but that's different than describing them as part of the "Third World" which involves more than just those issues.
It took me a moment to realize that you're probably not referring to the Ethiopian food on Telegraph Ave.