When and why did the people of the USA begin to believe their country to be a 'Christian Nation'?

I believe you are asking about the Christian right, which began in the 1940's and grew influential in the 1970's. The evidence for their argument that America is a Christian nation goes back to the Puritans.

It was never meant to be a Christian Nation, and the constitution only says "Creator" instead of "God". I know this very well. I'm merely wondering when they started to believe it, and why so many are trying to push this idea as fact.

Again, evidence for the argument that America is a Christian nation goes back to the Puritans. They try to push the idea as fact for the exact same reason anyone pushes their ideas as fact; by doing so you can make others perceive it as the truth.