I'm familiar with the phrase from the US Code of Conduct, which outlines how military servicemen should resist capture, attempt escape, and set up a chain of command among POWs. The first article begins: "I am an American fighting in the forces which guard my country and our way of life."

The Code of Conduct was instituted in the 1950s and was part of a larger social movement to identify America's conflicts with defining a certain way of life. One of my favorite documents from the era is the 1964 Report of the Commission on the Humanities. The report itself is only 16 pages long (you can skim it and ignore all the appendices), but it quite clearly defines what the authors saw valuable about an American way of life. In particular, they saw freedom of expression (arts) and freedom of inquiry (humanities) as two foundational rights of a democratic society, and they urged the government to remember that it was useless to win the Cold War if the US abandoned its culture along the way.

The phrase "way of life" is tremendously vague, but it peaked in use during the height of the Cold War, and I suspect that this shaped the meanings that the phrase still carries today. In this context, it's probably best to think of an American way of life is capitalist, democratic, and free; it is not socialist, despotic, or policed by the state.

With such a broad definition in mind, it's easy to see how almost any American conflict could be justified as an effort to "defend our way of life," and in doing so, how that shapes these conflicts. (For example, the transition in purpose from ousting Saddam and hunting down WMDs to building a democratic state in Iraq.)