Monica might have been lying to make Fine feel better, though. She did fake-cry to win Fine's sympathy (not in a mean way, of course), and unlike Fine, leaving the school destroyed Monica.
I read again the two chapters. Since, I wrote the comment after reading chap. 2 and edited it after reading the comments here. I didn't comment after reading chap. 1 because of the afterword that the essence of this story is in chap. 2. True to what the author said it metaphored the Catholic faith about heaven, hell, purgatory and wandering souls. As much as possible, I don't touch about religion.
So, going back, I've read the two chapters again for I might have missed something. I'm sorry I can't see the fake cry. What I saw is a child who was happy to finally have a room mate/playmate and was suddenly saddened when Finè signify she won't sing during graduation. The singing symbolizes your joy and happiness for the graduate. But when Finè explained, Monica suddenly smiled and was happy again.
To me, Monica's heart is a heart of a child. A heart that genuinely showed their feelings, doesn't hold grudges, easily forgives, easily trusts, doesn't judge, always sees the possibility because they live in the moment. When you make a promise to a child, make it happen for they will always remember. Monica was not destroyed by Finè leaving. She finally understood when they were running and her flesh was withering that she was "physically dying" and it was too late for her unlike Finè. So, the bidding of goodbyes was not to destroy but to reaffirm a promise - a hope was build to look forward to.