Forgotten Realms: Waterdeep. I'm rather surprised there's any debate over this. FR has a ton of cities, but Waterdeep is the city, no question.
I agree with you, though I can imagine a few old-school folks naming Suzail, as the capital of Cormyr (i.e.: where many campaigns were actually run in the early Realms).
Ravenloft: The Village of Barovia. Normally I'd consider Barovia to be more of a Pastoral Starting Town, but let's be honest, anything outside of Barovia is kind of an afterthought for Ravenloft.
Disagree when it comes to being the 'big city' in Ravenloft, regardless of how you define Ravenloft.
If you define it by the campaign setting (and the inclusion of Ravenloft on a list of other settings suggests you want to), then Jester David's suggestion of Paridon is a good one -- it's basically a city-domain with an attached subterranean domain, each of which has its own darklord. Much of the Core isn't really defined by the cities of the Core, but the closest to an iconic city in the Core is probably Il Aluk (which became the pocket domain of Necropolis for a time) in Darkon, also pointed out by Jester David.
If you're talking Ravenloft as in the original I6 and as re-interpreted in adventures like Expedition to Castle Ravenloft and Curse of Strahd, then the Village of Barovia is probably the best known location, but Vallaki is a larger town (roughly 1600 compared to just over 500 in the village) with a lot more going on, due to its position on the Old Svalich Road, the major trade route through the Southern Core.
Just my $0.02US.