I like to name the website project .Site so it is clear rather than .Web because sometimes people use .Web for a supporting class library.
Then if it’s a v9 project I’ll probably keep it all in the one project but if it’s v7 or v8 I’ll have a .Core project which has the services, models and business logic etc.
If I’m using separate projects I like to move my models builder models to a Models/Generated folder in the .Core project or in a separate .Models project. I don’t change the namespace of my models builder models though any more just to keep it simple.
Hope that helps mate