You're missing the central point in my opinion: citizenship, which is verry different from population. These were the verry core, the soul if you wil, of the city state. In most city states they were the, hereditary, economic and political drivers af the state and constituted the army (Greece: hoplites, (early)Republican Rome: legionaries.)
Territory, population and economy are secondary to this in defining a city state. Luxembourg for example is nothing more than a smal nation-state, not a city state.
A city-state could be a republic (Rome, Athens,...), a monarchy (Sparta), a tirrany (Syracuse),...
A city state would try to extend its influence over the neighbouring country (most of these lands would be owned by its citizens) and over the surrounding city-states by means of treaties, be they mutally agreed or forcefully imposed. When we talk about the Athenian Empire, we are actually referring to the Attic coalition: several city states entering in a treatie relationship, at first dominated by Athens, later dictated by Athens.
The first real city-state growing into something of an empire was Rome. Rome not only exerted its influence by means of military power or treatie, but also by extending Latin or Roman citizenship upon others, cutting the historical link between citizenship and the actual city.
Without the central notion of citizens, with historical rights and duties, you can't really speak of a ctiy-state, but of a small country centrerd around a city.
(Hope this makes sense, English is not my native language

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