There's some useful information in your description, but that's just a very vague description of how neural networks with sigmoid activation functions are trained.
Moreover, there are many other AI systems apart from neural networks (such as support vector machines, expert systems, etc.), which, of course, I cannot exhaustively list here.
Is my understanding of how AI works correct?
I would say it's not completely incorrect, but, as I said, it's a very vague description and it only refers to a subset of techniques in the AI field. With that description, no newbie would probably understand how neural networks are really trained, apart from knowing that you will train them iteratively and there are sigmoids involved.