Giuoco piano, does black have the advantage here?

b4 may be the most "natural" move, but it is far from the best. My first instinct would be 1.Qb3, but I have no idea what theory says. In the final position, after Black plays 4...d5, what move do you like for White? My non-master view is that White's extended queen side is week and he's going to have a hard time keeping it together. And Black has those bishops.  I think Black is going to be able to open lines and get a lot of piece activity. Why don't you take white and play a few times against your chess engine. Obviously, the engine is much stronger than we are, but it will give you a better feel for what Black's compensation is like

I won't argue with you, but I sincerely suggest you try taking the white pieces against Fritz or Stockfish or Komodo a few times. I think you will find it very instructive

BTW: I'm not suggesting that Black has a big advantage in that position, just more than enough compensation for the pawn

The engine gives me this continuation

So black manages to castle, get the pawn back and a much better position but this only happens if black does the best moves

From whites perspective the issue here is that finding the "best" moves isn't all that hard for black. Whites position after 7. b5 is already full of weaknesses, and the pawn structure is cracking. After 9. Nxc4 whites pawns are too far advanced, he has only one piece developed and hasn't castled. Black has no weaknesses, 2 pieces out and is already castled. That's 9 moves in with white pieces, where you have one pawn for bad position and you're behind in development. After that all you need to do as black is to start chiseling that central mass, get your pieces out and enjoy the initiative.