The magic list shows all of the magichip’s spells, even the ones that aren’t unlocked yet. The ones that are still locked are grayed out. When the chip levels up, one of the grayed-out spells will become available. Each chip levels up separately, so leveling up an ice chip won’t unlock new spells when switching back to the fire chip, leveling up a fire chip with higher-level spells won’t unlock anything for lower-tier chips, and so on.
I’ll admit it’s not the most intuitive system and there’ll be some changes in the next big update, but to my knowledge it should be working as intended.
I’d have to look at the code, but I’m pretty sure the Flame spell should be unlocked the second time you level up the fire chip.