Yes, it is wrong to think that the bombings were the first move in the Cold War, because the United States could not enter into a Cold War before it stopped the live one. World War Two was ongoing. Millions of people were needlessly killed, and the war was expanding. The United States acted to end the war, not start the next one.
The bombings of Hiroshima and Nagasaki are as good of a point as any to define as the start of the Cold War. The United States now possessed a weapon greater than any the world have ever seen before. To say we scared the daylights out of the Soviets would be an understatement, and they certainly spent the rest of their existence trying to keep up with our weaponry developments.
The bombings were not necessarily the first act of the Cold War although it is understandable how they can be come to be interpreted that way - as a show of nuclear power. At the time, the Communist threat was not at the forefront of the American government's minds so much as ending the campaign of the Axis. It is neither the first move nor the catalyst, although the shadow of nuclear power lingered over the duration of the Cold War.
It is not wrong to think that the bombings were the first move in the Cold War. We needed the Russians to understand that we were not afraid to make the first move and use this weapon against them. It was either that or we would have gotten attacked first.
It is not wrong to think that the bombings upon Hiroshima and Nagasaki were the first move in the Cold War. This is because the Cold War was started because of the uprisal of nuclear terror. This long, long war was definitely started by Hiroshima and Nagasaki and their bombings.
Truman decided to drop the bombs in part to intimidate Stalin. He wanted the Russians to know that America was not afraid to use nuclear weapons as a weapon of war. America also wanted to end the war with Japan quickly before the Russians could occupy the northern Japanese island of Hokkaido.
We did make the first move towards a cold war by using the atomic bomb. We used the bomb before it became fully known and popular. In today's world we would never have gotten away with the use of the bomb in the same way we used it back then.