I don't care if it's a web interface. I basically wish that Thunderbird's contact system was actually flexible and well-designed, then I'd be happy.

I want to manage hundreds of contacts and be able to tag them well so it is easy to focus on relevant contacts for various purposes. And I want a nice interface, easy to make notes and to add any arbitrary amount of emails, relationships, websites etc.

Google Contacts is closest but fundamentally compromises my control and privacy. Otherwise, Thunderbird, Kontact, and Owncloud all have mixed pros and cons where none is quite acceptable.