A Cyclone Phrase is one that either
contains two words which both have the same letter in the same position (e.g. "drug and dose" or "chug our ale")
or
contains a word in which the same letter appears the same number of places from both the start and the end (e.g. "blurb in my blog", "an astute moron")
or both. The one error is
"fork, cup, and dish" is not a Cyclone Phrase.
I got to this by wondering why it was called a Cyclone Phrase and looking for some kind of circularity or cyclicity in the letters, which led me to the ideas of alliteration and symmetry. Most of the examples have either two alliterative words ("all alone at noon") or a word starting and ending with the same letter ("cenozoic deluge"). From there, finding the actual pattern wasn't too hard.