The urban area of Paris ("aire d'attraction de Paris") extends well past the areas where people would consider that they are "in Paris".
It might well still be the biggest in the EU if you drew limits based on what people considered part of the city - I don't know - but it certainly would not be as big as its urban area.
In fact, I'd argue you're unlikely to find any major city anywhere in the world where most people would agree that every part of the outer boundaries of the urban area are part of "the city" (or even "a city"; parts of urban areas will still often be considered fairly rural by those who live there or in the nearby city) even in a loose colloquial or cultural sense. Even coming close would be exceedingly rare. This because the ways urban areas are designated by design tends to include commuter regions far outside, with their own identities, and often very separated from the biggest city in the urban area.
So while going by city limits will be misleading, so will going by urban area. Unfortunately, no single metric will be accurate for these things.
Just to point that "aire d'attraction" is not the same as the urban area. It is the area of influence and extends much further than the urban area.
I live outside of London, howver you define 'London', but still in London's area of influence considering how many people commute into London from here every day.
It's a tricky one, because by some definitions it fits what is often called a metropolitan area, but there's no formal, objective definition of either urban or metropolitan area that is universally accepted. You're right it's probably wider than most uses of urban areas in English.
At the same time it's specifically meant to be aligned to OECD and Eurostats definition of a Functional Urban Area, which is a definition meant to ensure comparable statistics across countries, which neither the "old style" urban nor metropolitan area terms provide...
Which really just goes back to the main point that you can get pretty much whichever result you want here unless you narrow it down to the specific criteria that actually matter to you...
It might well still be the biggest in the EU if you drew limits based on what people considered part of the city - I don't know - but it certainly would not be as big as its urban area.
In fact, I'd argue you're unlikely to find any major city anywhere in the world where most people would agree that every part of the outer boundaries of the urban area are part of "the city" (or even "a city"; parts of urban areas will still often be considered fairly rural by those who live there or in the nearby city) even in a loose colloquial or cultural sense. Even coming close would be exceedingly rare. This because the ways urban areas are designated by design tends to include commuter regions far outside, with their own identities, and often very separated from the biggest city in the urban area.
So while going by city limits will be misleading, so will going by urban area. Unfortunately, no single metric will be accurate for these things.