Willy Loman's version of the American dream is one in which athletic success, popularity, financial success, and being well-known and respected in a community loom large. House ownership, having a successful nuclear family, and an illusion of being "independent" or "one's own man" are also part of this ideal. This is very much an ideal still promulgated by many conservative think tanks, and used as a trope in political advertising.
It is not so much that Willy Lomax has an ideal "American dream" differing from that found in much of United States popular culture, but rather that he (1) fails to achieve it and (2) acts in ethically dubious ways in his quest for it. In some ways, the failure of WIlly Lomax can be read as a critique of the concept of the American Dream, arguing that it represents a sort of capitalistic materialism or "false ideology" that corrupts the moral character of the people who strive for it.