Hey Bobzinhha District is an official classification of an area, as in for legal or administrative purposes whereas neighbourhood for me can be the street you live on, the streets around you, your village or town etc. In other words it's not actually definable by size, it's based more on the sense of belonging to an area and knowing the people around you.
The only thing that i can think of it that a neighbourhood is a lot smaller area than a distric. When you think of a district in France for instance, that is a very large are, or the Lake District in Britain, this is a very large area as well.
I would confirm that district is large. You would never find a district in a village, whereas a neighbourhood would be a small part of the village which surrounds a defined point which could be your house or a school etc.
There may be an old thread which discusses districts, quarters, areas, neighbourhoods.
Found a thread: cantons, arrondissements, communes
In AE, a neighborhood is the area where you live, and where you have at least a passing acquaintence with your fellow residents. A district is a designation used only by municipal authorities, such as those for water or sanitation (rubbish removal, sewers, etc).
In the States, each state is divided into districts, thus deciding, for instance, how many U. S. representatives a particular state will have. So if you want to contact your U.S. Representative, you need to find out, based on your home address xxxxxxx, what district you´re in. Neighborhood refers to different "xxxx" in the town/city you live in. For instance, I live in Timberline (neighborhood), in the city of Covington.
Mod edit: please note that this is the English only forum. Sprinkling other languages around is not required, as you have addressed the question perfectly well in EN.
In dictionaries and in most comments here in the forum, natives say that "neighborhood" is the area where someone lives, so it's linked to a sense of belonging. However, I noticed that "neighborhood" is sometimes used to refer to an area of a city, even when we're not referring to a certain person.
For example, if two friends are talking about an area of a city they don't live in, which word would they choose: "district" or 'neighborhood"?
I think Xxxx is a crowded neighborhood/district.
Is my observation valid?
And if it is, does this mean that neighborhood and district are interchangeable in this context?