What does the American Dream truly mean?

The American Dream only always ever meant having a republic with no nobility and thus no caste system. In short, the American Dream meant something like your life is what you make of it because of the design of the USA, the rules, should favor no one.

By having a republic with no caste system, no nobility, anyone with thrift and industry who serves enough of his fellow Americans can get rich. Because of the republican design, the only rules that would exist would be those that create a level playing field so that anyone could achieve in direct proportion to the effort that one would put forth.

For years now, many have engaged waged a war of propaganda to convince Americans of a different meaning for the American dream. They have tricked too many Americans into believing the American Dream means equality and by equality they mean equal outcomes. For the tricksters who favor a regulated economy, they would like everyone to have the same income and debt with themselves having the rewards for controlling the rules. For the tricksters who favor Marxism, they would like everyone to live under socialism with themselves having the rewards of privilege.

In a 1931 work titled Epic of America, James Truslow Adams captured the essence of the American dream — your life is what you make of it because the rules should prevent the existence of nobility and a caste system:

But there has been also the American dream, that dream of a land in which life should be better and richer and fuller for every man, with opportunity for each according to his ability or achievement. It is a difficult dream for the European upper classes to interpret adequately, and too many of us ourselves have grown weary and mistrustful of it. It is not a dream of motor cars and high wages merely, but a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, and be recognized by others for what they are, regardless of the fortuitous circumstances of birth or position.
From the 57th Congress of 1901 to the 115th Congress, the one that began in 2017, there have been 59. Each has worked ever more to wreck the original design of the USA, the design again nobility and caste. 

It would take a team and their lifetimes to catalog every action by congresses since the 57th which have contributed to the demise of the republic. Of course, too, states legislatures were complicit in the demise of the republic with the passages of the 16th and 17th amendments.

Suffice to say, with the creation of a slew of welfare (TANF, SNAP, Section 8, EITC, etc.), subsidy to jokers like Elon Musk and his Tesla, and bailouts for the rich when they decide horribly wrong (2008 Banking Crisis, major banks, GM, etc), successive congresses have created a caste system. Along the way, successive congresses managed to create an Establishment's media that has worked wonders to create personas of nobility.

All games have rules and thus all games have design. The outcomes of all games get constrained by the rules.

Successive congresses have changed the rules. Their rules changes have created a caste system and nobility. Their rules changes have created an American nightmare for almost all but an American Dream for those at the top and the corruptible lazy at the bottom, including illegal aliens.