IMO "design design" is the ability to look at, say, combat system and point out flaws and how to improve them. Still, such ability is limited what can be coded in realistic time frame and your sense of what's cool and fun. Not really something you can teach in the general sense, there is no "general theory of game design" you can apply. You just know how to change something boring in the game to be fun and how to code that, you learn that from knowing how to program and from being a fan of the genree and playing the game in question a lot.Designs have to be broken down in order to be implemented. That's what differentiates designers from idea people, who can only hurl out scribbles on a whiteboard/post-its while daydreaming on a chair of how they'll make a name for themselves.
I think X is cool!
Ok, but we need Y and Z in order to put X in.
A designer would be able to break that down if what they came up with has a solid foundation. An idea person would just say "Make it work, I'm just the designer". Fuck off, I expected a whole design document out of that, not a single line written on your notepad during lunch and stained with mayo from the deli next door.
I have no experience with game design schools, I guess they're either blatant scams or skinner box-tutorials.