To your mind, what differentiates hamlets, villages, towns, cities etc?

Hamlet: I never use the word but if someone used it I would think of a collection of houses, probably without any public businesses.

Village: A collection of houses, with at least one public business (bar, gas station, etc.) And probably at least 1 stop light.

Town: Despite growing up in upstate NY I don’t think of the phrase as meaning a Township. I think of a town as a large village or small city. If it doesn’t have a downtown (i.e. a dense collection of public businesses, preferably of multiple stories,) then it isn’t a town. Not even a “small town”.

City:definitely has a dense collection of public businesses, this time they have to be multiple stories (the buildings, that is, not the businesses themselves.) The larger the downtown, the more likely I am to describe it as a city and not a town. If it has its own organization for public transportation, then it is definitively a city and not a town.

American here. My definitions, plus a few more:

Hamlet

The melancholy prince of Denmark, who spends way too long making up his mind about suicide

or

The smallest of settlements, almost certainly not incorporated except possibly as a joke. Probably just four farm families who decided to build their farmhouses on the adjacent corners of their lots. 100% agricultural.

Village

Larger than a hamlet, and probably (though not necessarily) officially incorporated. Has a few businesses to serve neighboring farmers (at the very least, a post office, a store, an elementary school, and a church), but the dominant industry is still agriculture.

or

A 'burb, development, retirement home, or shopping center that’s trying to sound quaint.

Crossroads

A small settlement at the intersection of two significant through roads, primarily catering to travelers. Has, at the very least, a gas station and a restaurant (probably several), plus possibly a motel or three. Some overlap between this category and villages.

Town:

Larger than a village, and has some major industry besides agriculture, though probably only one employer in that industry. Likely to be the county seat. Has all manner of other businesses, though probably a small selection of each. Probably has a high school/school district. May be further qualified based on the type of industry, such as a college town, mining town, etc.

or

Colloquially, any municipality, regardless of size. One might speak, for instance, of one’s hometown, or going in to town, or the like.

City

Multiple industries, with multiple major employers. Has multiple zip codes/post offices. Has schools at all levels, and its own public school system. Multiple distinct neighborhoods.

Suburb

A municipality economically tied to and contiguous to a larger city (that is, it either borders the larger city directly, or it borders another suburb of the city).

Metropolis

A city with suburbs. Tend to be large, but not necessarily.

Bedroom Community

A suburb which was built deliberately as a suburb. Has no significant industries of its own; most residents work in the metropolis or in one of the other suburbs.

The 'burbs

Bedroom communities composed entirely of planned developments, rather than neighborhoods. Houses all look the same, and are generally under the sway of homeowners’ associations.

A three tier system also seems to be present no only between settlement type but also some of the facilities expected within each. For example, education is typically split into primary, secondary and tertiary while health care is also split into primary care (family physician), secondary care (specialists like cardiologists and dermatologists) and tertiary care (surgery, severe burns). Economic sector are also split into three, typically, with agriculture/fishing/hunting/mining being typically of villages, some manufacturing often being found in towns and cities being typically overwhelmingly service-based.

A quaternary sector is also sometimes present in education/healthcare/economic classification. Can one argue the same for settlements? If so, what would that settlement type be? What would be typical of it in terms of size, facilities, activities etc?

To my mind, city means a large metropolis with a diversity of population and cultures. The line between town and city has to do with the cosmopolitan feel to a city that a town doesn’t have. I know of towns with 6-digit populations and cities with 5-digit ones.

Hamlets, villages and other smallish communities (including suburbs and neighborhoods) just don’t have much going for their identity than their location away from bigger places. Little is the keyword, and quaintness is the subdivider of those little places.

Not much help perhaps, but that’s how I think of the issue.

Most of these terms are not in common use in my neck of the woods. I would personally define “city” as “any municipal region which is incorporated as a government entity” and would refer to any other inhabited place as a town or neighborhood.

I would also use “town” to refer to a small city.