District has many meanings besides those you mention. Just where I live there are a Fire Prevention District, a Sewer District, a Solid Waste Management District,, a School District (with Football and Basketball Districts), a Special School District, a County Council District, a Municipal District, a Missouri House District, a Missouri Senatorial District, a Federal Congressional District, a County Library District, both State and Federal Judicial Districts, a Federal Reserve District, several distinct Licensing Districts, and lord-knows-how-many more defined by other State, Federal and local regulatory agencies and departments. Most cities have one or more shopping districts and business districts and entertainment districts and other "districts" of a purely commercial or architectural or other non-administrative character, some quite famous: the Design District in San Francisco, the Garden District in New Orleans, the Clothing and Financial Districts in New York. And churches, unions, and other associations organize themselves into districts: my son spent eight years as a Cub/Boy Scout in the Pathfinder District.
And the seat of our national government is the District of Columbia.
So I'm afraid the question is essentially meaningless. A "native speaker" (or a non-native speaker, for that matter) is not except in very artificial circumstances called upon to exercise her "intuition" about a word's meaning outside a context. Tell me what we're talking about, and I can tell you what pops into my mind when you say district; without a context, it just means an area bigger than the block I live on.