Belle’s story also came after Rumple’s story, so it can also be another explaination of why it is not in the book. Rumple never told Belle much about himself. He said he “had a son and then he was gone, end of story.” She can’t write down what she does not know right?
But Rumpel’s entire story has been left out. His part in the curse. Him helping the queen. His name isn’t even in the book. It almost seems intentional.