Here's an update: Since I started offering Luna some kibbles of dry food in the early hours of the morning (before breakfast), she didn't throw up anymore - until today, when I forget to give her her "pre-breakfast meal". So I'm pretty sure it's this stomach acid thingy that makes her sick when she's hungry. I have also asked my vet about it, and she said that some cats have these problems when their stomach is empty.

Lily, btw, is more that happy about the additional treats in the morning. Whenever I reach for the drawer of my bedside table (where I'm storing the food), she would get up and start begging. lol