The early republic, Washington through Monroe made us who we are. Post WW2 era Truman through Kennedy might be the most "defining" part of American history.
Those aren't events.
For the first span of time you listed, the most notable events are the Louisiana Purchase (part of Westward Expansion), War of 1812 (attempted expansion which ended up not changing the status quo), and some Indian Wars.
The second span of time you mention is marked by the end of WWII, the Cold War, the Space Race, and the Civil Rights movement.
So which of these is the most defining part of American history to you?