I am a piano beginner and am learning by myself. I'm very confused about the upper notes (original sheet), and I rewrite them to become the lower notes (rewrite sheet). My question is that are they correct?
Yes, the rewrite is correct.
The original sheet music is an example of "cross-staff beaming", and its purpose is to help keep visually intact related groups of notes. In this case, the intention is that those note groupings be shared across two hands, and the rewrite shows this.
Sometimes cross-staff beaming can also be used because one hand plays notes high or low enough that many ledger lines would be needed. Instead of that, cross-staff beaming can be used for a single hand just to make the reading easier.
The original notation shows the musical sense AND gives a reasonable suggestion of how to split the notes between the hands.
Your version makes the same suggestion for which hand does what, but removes information about the musical phrasing. It adds nothing, and takes something away. So it's inferior. Don't do that!