This is C1 or C2 level so if you cannot understand this answer don't worry about the difference.
Zone: less commonly used in general English. Military examples: "exclusion zone" "DMZ", "green zone". Also used in science e.g. subduction zone. Also sometimes used for a state of mind e.g. "in the zone".
Region examples: Great Lakes region, subtropical regions.
Area: very commonly used word e.g. area = length X breadth
District: Smaller than a region. I am not American but I think this has a particular meaning in US English e.g. congressional district, Washington DC