What's the difference with when I add a beatmap as a favourite in the site and the ones that show after you play, namely: Favourites A, Favourites B, and Online Favourite?
I'm guessing favorites in game are lists of songs you want. You may use the first fav for your favorite songs and the second fav for beatmaps just outside your scope of completion you want to come back to complete.
And then online favs I think are just a way of liking them/for them to show how liked they are.
Wrong forum, Favorites A & B lists are local only, and you can filter the maps shown in-game by Favorites A or B, Online Favorites is the same thing as clicking "Add to Favorites" on the site.