I'll have to see if there is a way to change what shows up in that upper corner. You might try in the upper corner clicking on the one that looks like bars instead of squares as that groups them differently and might solve your problem. When in the squares view it does do it alphabetically. I'm somewhat limited by the Pivot Collection itself in what I can do. For example, it only allows the "or" sorting, not "and" in each filter category, which is why I made 3 different type filters.
I'm not sure what you mean in the last couple sentences about what you want the type search to do. The way the pivot program does each filter is say you click in the Main Type Fire, you get all the main type Fire ones. But if you also click Water, you get all the main type Fire ones, and all the main type Water ones. I added the 3rd Either Type filter because say you wanted a Steel type, but didn't care if it was main or secondary, if you do Steel in main, you can't see the ones that are Steel as secondary because it filters them out, so you'd have to do each search separate and that isn't too useful.