President Truman's press release about the atomic bomb says nothing about whether dropping the bomb caused the Cold War. It could not have said anything about it because the Cold War had not really begun at that point.
You can argue that the bomb helped to start the Cold War because it made the Soviets afraid of the US. You can argue that one reason for them to be so antagonistic towards the US was that they were afraid the US was going to use its atomic weapons to try to control the world. In this way of thinking, the presence of the atomic bomb made the world seem more dangerous to the Soviets and made them more likely to be distrustful of the West.