Donovan playing for the Galaxy is a bit different than he is with other teams, and it actually reflects a bit of his style during his last stint with Everton.
Moyes was smart in the way he deployed Donovan - he assigned him a relatively simple role, sticking to the wing, primarily opening up space for his teammates to operate in, but available for a pass as well. Even when he did get the ball, it was mostly just a run down the line and a cross that was usually just okay, not great. He was fairly conscientious about tracking back and defending, with a couple errors in that role early in his time there.
At the Galaxy and for the US he has always been much more important - since he is usually the best or close to the best player on the team, he has been tasked with much more to do.
But ever since playing for Everton, and with Bruce Arena taking over the mantle at LA, Donovan has been much more of a player that is simply determined to win. Usually, that means winning ugly. He's not about flair, and he's barely even about scoring goals. He's about trying to keep possession, and pressing the opponent. The reason you don't see him do as much is he just doesn't stray into the opponent's penalty box all that often anymore, really just a few times each game.
That isn't to say he won't be more of an attacking player for you guys - in fact, I assume he will be. It's just a totally different situation.