Yes that is what I am saying, the order is different: "A implies B" equivalent to "not B implies not A" but not "not A implies not B". The order is critical. And that would be P -> Q <=> (~Q -> ~P) as can be seen on Wikipedia too.
Going back to the rain example, not raining does not mean not cloudy, but not cloudy means not raining. Just like you're saying: snowing (not raining) means cloudy which means not raining -> not cloudy is not correct. So I would think it is:
raining -> cloudy
not cloudy -> not raining

I like these logic questions!