Also of note is that 손목시계 is actually, more accurately, "wristwatch" instead of simply "watch". I don't know if "wristwatch" isn't accepted as an answer or not but that I found that good to point out
Yeah, it's not accepted as of now. Really weird. I went ahead and flagged it.
saying 'putting a watch on with one hand' is more colloquial and natural than 'putting on a watch with one hand'