This is really confusing. What do these words actually correspond to?
A neighbourhood or a suburb = a specific area, not necessarily on the edges of the city. The suburbs = outlying areas on the edges of the city. A suburb = an outlying area on the edge of the city.
So if "we live in the suburb" is correct, "We live in the neighbourhood" should also be correct.
If "we live in the suburbs", is accepted then neither of the above should be accepted.
In Canadian, American and Australian English, it is incorrect to say "I live in the suburb" to mean "I live in an outlying area of the city". "I live in a suburb" would be ok though.
Can someone fluent in French possibly answer this?
La banlieue is, as you say, a collective noun to refer to the outlying areas on the edges of the city.
"Je vis en banlieue, dans la banlieue de Paris, dans une banlieue sympa" are variants to mean that, depending on context.
It is also administrative, since these areas are divided in cities (or villages) with their own administration. Since the cost of a square meter will decrease as you move away from the main city itself, the closer banlieues will be rather urban with tightly packed buildings and further banlieues more often consisting of spaced houses with gardens.
"Les banlieues" refers to either different areas around the same city (les banlieues nord et sud de Paris) or all areas around various cities (les banlieues des grandes villes posent souvent des problèmes de violence, liés au trafic de drogue et au chômage).
OK. So, in English, we don't use "the suburb" or "the neighbourhood" as a collective noun, so this translation is inaccurate. We'd use "the suburbs" instead, but that would also mean the same as "Les banlieues". But if I said "I live in the neighbourhood" you mean in the area of where you are standing, or the area of somewhere you just mentioned, like "Parkdale is so nice, I live in the neighbourhood". Neighbourhood has no correlation with the "outlying area". In Australia, a suburb is like an official neighbourhood, and has no "outlying area" correlation.
Suburb, in US and Canadian English, does mean outlying area. In Canada, you might say "I live in Oakville, a suburb of Toronto" (meaning outlying area of Toronto). But you'd never say "I live in the suburb." and just leave it at that. That's incorrect. You'd either say "I live in a suburb" or "I live in the suburbs".
In American English, a suburb or the suburbs, always and only means an area just outside of a city. If used differently, it's being used incorrectly. A neighborhood can be anywhere and refers to any particular area where people live around one another.
That said, I live in the suburbs would be a common phrase, whereas, I live in a suburb would be said very rarely, in the context of a conversation. For example: What kind of area do you live in? Oh, I live in a suburb.
CBD : Central Business District which is, strangely, the heart of the city comprising Big Business, big department stores, boutiques both High End and low, and hotels. Everwhere else is measured from what is or was called the GPO : The General Post Office. It seems archaic. Spreading out from there are the suburbs, inner then outer which make up the metropolitan area. Further afield is 'the country', then rural areas leading to other cities and finally 'the bush' and remote areas. Well you can tell it is 4AM and I'm feeling chatty.