District is the neutral word
One name covers a US county, a French department, and a UK district, because district was already the word for that layer in the most countries.
Every country picked its own word for this layer of government generations before an API needed one.
The US calls it a county. France calls it a department. The UK calls it a district. All three are the same administrative rung, one step under a state or region, and this API needed exactly one field name to mean that rung everywhere. county was the first instinct and the wrong one, it is a US-specific word being asked to describe a French department, which nobody in Paris calls a county. fips names an American encoding scheme, not a place. admin2 is the correct GIS term and also jargon nobody outside a GIS team has ever said out loud.
district won because it is already the neutral international English for that layer, the word India, Pakistan, Kenya, and Portugal reach for on their own. The local flavor still shows up, just one field over, in type: county, department, district, whatever the source country actually calls itself.
Try it
Call /district/{code} on a US FIPS code, a French INSEE number, and a UK GSS code. Same field name every time. Check type for what the country actually calls it.