Ano ..., the back-home scene... Satoshi gives Naomi his piece of paper. If using a piece that is not yours screws you up, it is also Naomi who shouldn't be transported. If so, the only explanation that Naomi could use his and he couldn't use Yuka's is that he gave Naomi his, but he has a random piece paper (probably torn off the diary) and the piece of paper is unrelated to their charm-circle. That's why only the part b/w Naomi and the class rep worked, but broke b/w the girls and Satoshi.
Because it wasn't her piece of paper, but a piece of paper from this charm spell (regardless whose exactly), Naomi managed to get back, but at the price of being obsessed with the girl, whose piece she took?