How are you differentiating between ward and district (phuong / Quan)
Is "area" a bad translation here?
Area would be vùng or khu vực. As a rule of thumb:
town = thành thị, thành phố
city = thành phố, đô thị
capital city = thủ đô
area = vùng, khu vực
suburb = khu vực
district = quận
ward = phường
Some cities divide themselves up into smaller "cities" which are then divided into "suburbs". This can be seen in many western countries like the US and Australia. Other places like in France or Vietnam, cities are often divided into "districts" (similar to smaller cities) and "wards" (similar to suburbs). If we go further (many rural areas) will divide themselves into ấp (hamlet, think of villages) and xã (a commune, community of villages).
I am also greatly wondering about your "districts." Are they like the parish districts of New Orleans? Out west, we say areas sometimes, but most commonly just neighborhoods. Are your districts different than our neighborhoods? Like, we might say "the industrial district" (but usually the industrial area) but other than that, districts signify political polling areas.