You are totally correct: cycling is not draining mana in the sense of the CotC definition!
According to the support page https://d3go.helpshift.com/a/magic-puzzle-quest/?s=gameplay-questions—general&f=mtgpq-abilities-gallery&p=all cycling is defined as follows:
Cycling

When this card is exiled from your hand, if you have at least X mana accumulated in your cards, where X is the amount listed next to the keyword, and you are drained of X mana. You then draw 1 card.
(I have highlighted the word: drained)
Furthermore there are 3 definitions relating to the Cycling mechanism, which muddles the picture. The definition above is most likely the definitive, but there is also a Cycle definition and a icon description to the cycling icon.

On top of that the “stored mana” icon states
Cards with stored mana can’t have their mana drained or raised.
(Taken from the Hydroid Krasis card)
Cycling a card will reduce the amount of stored mana, but will not trigger the ability of CotC.
(Tested with Hydroid Krasis and Ugin’s conjurant).
So the word “drain” is used inconsistently, which may not be all that surprising since it is two different developers who made the implementation of “drain”.
Oddly enough, when casting “Rhona’s last stand” CotC does trigger, so it seems that it is not a set specific issue. Update - but RLS does not drain cards with Stored mana!
To end, I would like to state that I do acknowledge the developers have a huge task trying to align such inconsistencies nor that I expect it to be fixed anytime soon, given the size of the development studio. I’m simply saying that it is these types of inconsistencies that makes MTGPQ such a surprising adventure to play, because you never know what you are going to find behind the next bend in the road.