Sarah sees the American Dream as having opportunity. For her, the American Dream is to have the ability to start from scratch and build her way up to success.
The American Dream (interview)
I can turn this ugly old warehouse into an amazing place, and even run a business
Gatsby's American Dream is to rekindle the history he had with Daisy. When carving out his future, he feels that it can be anything, even the pieced together remnants of his past.
Daisy will come back to me and we will happily live together because I live across from her now. We can make things as they were five years ago
Gatsby's Idealized American Dream
Gatsby's corrupted American Dream is that, when looking to improve his life and becoming successful, he falls in love with Daisy, but the initial attraction is mainly because of her money. He saw wealth in her eyes.
Oh my! Look at that beautiful RICH girl. I would love to date her.