Fixed now. 'Writing into this book" is not common usage. But does happen typically as part of a phrase. Perhaps you are exasperated with someone asking you to make sure you remember something and you reply, "I will remember it! Look! I am writing into this book, 'Must learn Hungarian every day!'. See? I've written it just there!"
(I don't recommend this stroppy tone of voice with your friends and loved ones ;-)
Those are different actions. If you're writing in a book, the book already exists, and you're writing on the pages in the book. If you're writing this book, that means you're the author, and you're in the process of creating the book.
"I am writing this book" would be Ezt a könyvet írom. Your sentence makes "this book" the direct object, whereas the other sentences (writing in this book, etc), do not.
I was almost finished Choices 3 Part 1, when my work disappeared. I am not in a position to start over. Please fix this. I was about to report on my answer: Szines parnak esnek le erre az agyra is. I believe it should have been accepted and it was not. Also I would like to continue where I left off or I will be unable to do more today.