Answer:
Neither, technically
Explanation:
In my opinion, it was actually Germany who started the Cold War. I know that sounds weird, but hear me out. After the end of World War II, the allied powers divided Germany and Berlin for themselves. Meanwhile the Soviets would establish Communist dictatorships in Eastern Europe. Germany was the first place that the powers engaged against, with the West uniting their parts of germany into the Republic of West Germany, while in the East, the Soviets created East Germany.
Germany was the real contester that both the Soviets and the West wanted. Now obviously this kind of proves that the USSR started the whole thing. Most people are biased against the USSR, and I can see their point. However the US was the first to back the proxy wars, with the Greek Civil War and the tensions in turkey, thus lighting the fire for the chaos to come
So who actually started it? Depends on who you ask.