Disclaimer: This is for anyone who does not understand certain rotations. If you do understand skiing, everything I say will remain quite obvious, and probably sound stupid...so who cares?
If anyone is really confused (as you seem to be), a "cork spin" originally a "d-spin" is essentially a backflip 720. People make their own variations and axis of this which, in reality, is what style is...that is why moguls and aerials are you know...conforming and freeride skiing is not conforming because people figured out how to use grabs and such to make tricks looks better and personalize them. The issue of the cork 3, flatspin, and rodeo is really not that hard to understand, youre making it too complicated. Flat spin: you are spinning flat do get so confused and make it difficult. The names of tricks are generally like that for a reason. The true definition of a flatspin would be spinning on an axis of a pole basically sticking sideways through your body (parallel to your wingspan, this pole would stick straight up out of the table, or knuckle or whatever into the sky). A rodeo is a convoluted definition of a flatspin. It brings us back to the style issue. Everyone does a rodeo differently (style) some are flatter, some spin and flip more, but they all generally the same. You pop sideways from the jump, this is why it is very difficult to do a true rodeo or flatspin on a trampoline (most people do variations of lincoln loops including me for the most part) because there is no jump. If you dont understand that you should ski more and watch more skiing until you do. A cork three can be similar to a rodeo in the way your body is in the air, but it is a fundamentally different trick. You see the similarities when a person does a double flip. A double cork 7 and doble flat can look very similar because there are few ways that are easiest to go into a second flip, and come out of the flip(s) (I can not do any dubs but i have watched enough to know this). The fundamental difference in the cork 3 and the flatspin is your edges. the way you take off and land. They are basically the opposite. If you dont understand that go skiing until you do, and if you dont understand after that, youre a kook, get over it...
This is convoluted and stupid, i dont know why i tried to explain this because it will confuse half of you and the other half will think i'm and idiot. But who cares, go skiing and stop worrying about what the trick is called because it doesn't matter in the scheme of things. Y'all get to caught up in what shit is called and not what it looks like and how much fun people are having.
Have a great day i hope you go skiing soon