Was the United States of America fighting for freedom or democracy in the Cold War?
There are two ways to read this question.
First, it could be read as “did the US fight for freedom and democracy (as opposed to something else) during the Cold War. If that is the question, we can argue that the US fought for freedom and democracy, but also for its own interests. In other words, freedom and democracy were not the only things that the US was fighting for. We can see that this is so, for example, from the fact that the US strongly supported regimes that did not necessarily offer their citizens much freedom or democracy. South Africa is one example of this. The US supported that regime even though it denied freedom and democracy to black people, who were the majority of the population. For this reason, we can say that the US was not only fighting for freedom and democracy.
Second, we can read this question as “did the US fight for freedom or did it fight for democracy?” If this is the question, the answer is that the two are essentially impossible to separate from one another. Fighting for freedom is essentially the same thing as fighting for democracy. The answer, then, would be that the US fought for both freedom and democracy.