The most important aspect of the Puritan colonists' understanding of themselves is the fact that they felt they were meant to create a "city on a hill" that would be an example for the rest of the world. This fact shows us that this "founding culture," at least, saw itself as superior to other cultures. It also shows that the Puritans felt that they were both able to and obliged to create a perfect society on earth.
There are at least two important results of this that can be seen in both the short and long terms of American history. First, there is a tendency to see our own culture as superior to that of others. That has, one can argue, led to such things as the eradication of most Native Americans and to much later efforts to spread democracy around the world. Second, there is an abiding belief in reform in the US. From some of the earliest days of our country, we have believed that we can arrange things so that our society and culture can become more and more perfect. This has informed such movements as the abolitionist movement and Prohibition.
In these ways Puritan colonists' understanding of themselves helps us understand more about the "founding culture" of our nation.