As far as I know, William Dembski is the person mainly saying things about how to tell what is and isn't design. His analysis uses the term "specified complexity". If you google for "William Dembski" or "specified complexity" you should be able to locate some of the literature.
I suspect your first picture would be consider unspecified complexity (the result of random processes), so would not be a candidate for design. I'm not sure of the snowflake, but I think he counts it as specified complexity, but rules it out as design since that specified complexity is, in effect, specified by known lawful processes.
As best I can tell, Dembski's writings are pretty much all philosophy, with little or no empirical work. That is to say, there does not appear to be any measuring procedure that you could apply to a real thing that would give you a "design probability" readout.
I see this as a "God of the gaps". That is, we infer design from our lack of knowledge as to how the complexity arose. But it is always possible that future science will explain that particular complexity (much as we can explain snowflakes), so the design inference is made based on our current lack of knowledge.
I think Dembski is basing this on Kolmogorov complexity. And that's what lets the snowflake out. A snowflake turns out to be not all that complex, since all of the symmetries allow a relatively simple description.
It's been a while since I last tried reading any of Dembski's work, so the above is from memory and might be a little confused.
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