Blue Burst changed the requirement of "beating dragon/worm/computer boss & talking to principal after". Now you have to do new 'governement' quests which are available from the principal's office in multiplayer mode from the new npc momoka (ovenmitts lady) and attempt to tie the plot together a little better. They have names like 1-1, 1-2, 1-3, 2-1, 2-2, etc. Having 2-1 available will allow Caves oneplayer quests to show up, need 3-1 for mines quests, and 4-1 for ruins quests. Once you unlock 4-1, then all the oneplayer quests unlock the same way you remember on previous versions.
Ep2 did not have any official oneplayer quests, so that category simplay has nothing in it. It's just there to separate it from custom oneplayer quests.
There is no guide to all to what all the custom quests are, but I encourage you to explore around. Many of the added quests are quite cool.