La cité de Paris n’est pas très étendue is not an idiomatic sentence. In the case of Paris, cité is more related to the old historic center as in l’île de la Cité.

We’d rather say, la ville de Paris n’est pas très étendue or le territoire de Paris intra-muros n’est pas très étendu.

Cité, as you already state in your question is either used to name an historical town (la cité des Papes) or its center (la cité de Carcassonne). Technically, the cité used to be the part of a town surrounded and protected by walls.

Cité can also be a neighborhood (grand ensemble) often but not necessarily poor or suburban (la cité radieuse, la cité universitaire). It is also used to name some large museums like la cité des sciences et de l’industrie, la cité de l’espace.

In conclusion, cité is indeed very rarely synonymous with ville.