British schools tend to fall in the ‘international private’ group along with the ‘international school of …’, though they tend to be smaller. They are fully private, very expensive, teach in English, follow the U.K. curriculum or maybe the IB in secondary. They are ideal for international, expat, mobile families who want their children to slot easily back into an English speaking country after a few years. Or high earners who want to move in these circles, but don’t necessarily want to integrate in the host country.
Catholic privées are French, and usually sous contrat to the state. This means that their teachers are funded by the govt and they have to follow the national French curriculum, but are able to have religious practices within the school too. In practice many non religious families use them - they often have smaller classes, more discipline, selective entry so are seen as superior by some. They aren’t that expensive, as they are state subsidised, and they expect quite a lot of family involvement. In my city, certain Catholic privées are very ‘establishment’: they are where the old elite families send their children , plus anyone who wants to buy into that status. They are seen as strict, rigorous (there’s no higher praise for a school in France 🙄) and very academic. if you want to establish yourself and your family in France, and follow the French way, this is where to aim for.
If you are heading to Paris, there is Saint Germain, with the British section. They are set up to take native English speakers who want to integrate with the French system and provide FLE support. Children work towards the Bac, plus extra work in their Section langue.