In my thinking Singel Crown was = to the first crown. But in the rules it says:"You are allowed to have and use any numbers of crowns." So are you allowd to make a 3rd Crown run with items you already found earlyer? This would change everything. And make everything alot faster.
So yeah, it was a little weird coming up with categories for this game but yes. Single Crown with unrestricted items means you could do it on a run where you've already eaten the pizza two times and fully stocked up on items. Nobody has done this yet.
I am intending to do it but right now my game has no items unlocked because I did all my no items runs, and I've been too lazy to go back in and unlock everything.
For runs limited to the actual first crown, those would be level runs for first crown.
Also, note that it's an "all squares are rectangles, not all rectangles are square" situation. Any one-crown level run on the first, second, or third crown would also be eligible for the single crown category, and the same run can be submitted for both if you so choose. The current record is for both single and first crown.