都市 in my opinion is closer to "main city". 城市 just means "city". 都会... no idea. I tend to see this as part of a name, rather than refer to "a large built-up area". Urban meeting place perhaps?
Some of the meanings get a bit muddied over time. Like Myeongdong, Insadong, etc. in Seoul. Why are there so many holes in a city?