In some ways, I think it's actually better to say that it was more than just the U.S. that won the Cold War, because I think it's important to remember that the Cold War was not merely a power struggle between the U.S. and the U.S.S.R., but rather a war between Freedom and oppression.
With all due respect, I think you are trying to put too fine a point on things. Of course there were many who worked to bring down the Soviet Union. But in looking at the big picture, it was not Poland that brought down the Soviet Union. It was the United States putting constant pressure on the Soviets for 40+ years. Obama’s intent was not to say what you are saying, but to diminish the idea that the U.S. was central to the defeat of Communism. IMHO