How do spells decide which intro to play?

Have you looked at bank C3 at all? That's where all the menu code is. There is an excellent disassembly from Novalia Spirit available on RHDN. I'd recommend searching banks C1 and C3 for references to $1D54, and then focus on instances that follow with AND #$07. That's where the menu setting for the magic order is stored in RAM.

There's a table at C3/4F49 that you will need to change. There is also code at C3/2C14 and C3/514D that will need to be modified with your new spell indices.

I'm surprised your changes to spell sorting didn't work in the battle menu. It looks like you've changed the right thing. I'm also surprised that Quick doesn't work. I don't see anything else to change other than what you've already done.