There are at least four different ways that we could define a city, and each would yield a very different geographic area. The archaeological definition yields something that looks like a city. Archaeologists would use these criteria to determine that a newly discovered ruin had once been a city as opposed to a rural village. Air travelers might use the same criteria to decide that what is visible on the earth below is a city.
Cities are also legal entities.
Cities as legal entities are normally smaller than the area that our archaeological definition would yield. Increasingly, the area that functions like a city on a daily basis is no longer a compact agglomeration of people and activities.