Such a city is still under two higher levels of government; it's hard to call that "self-rule".

Also, this kind of administrative definition of "city" causes weird conflicts. It's easy for a city, such as Los Angeles, to be much larger than the county it's nominally located in, such as Los Angeles county. Conceptually, it's also easy for a city to be its own county, which is a different status than "not contained in any county" -- but in practice where this is supposedly the case, the city will not actually match the county boundaries, instead being noticeably bigger or smaller.

I conclude that, in order to make the claim that cities usually have self-rule, the people who would like to make this claim adjust their definition of self-rule so that it covers the things they're already sure are cities, rather than having a definition of self-rule in mind and observing that, by apparent coincidence, most "cities" turn out to have it.

In the context of a city, self-rule means the place has a mayor and a city council. It can pass local ordinances, have a local sales tax, and provide services to the constituents.

Not all places start out with those things. The rules differ by state, but generally, you need a certain population, a percentage of whom must petition the higher authorities (which you mention), and then vote as a community to create the create the infrastructure and take care of themselves.

At that point, they lose a lot of the benefits provided by those higher levels, like the county and state service providers of fire, police, and garbage disposal for example.

Virginia has pre-emption of pretty much all matters of law. Cities and counties can only pass zoning ordinances.

For example, it's not possible for a VA city or county to make something a crime, unless there's a specific law on the state level allowing them to.