Wow.... the confusion!

Cool down timer starts ticking the moment it is triggered.
Reset happens every day at the same time.

Progression Cool Down primarily exists in games to slow player progress and force players to buy means to reduce them. There is no cool down reduction in this game (other than researching traits); the horse training cool down therefor serves no purpose.

Daily resets for everything where cool down can't be reduced.

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edit

Don't introduce new reduction scrolls either, sneaky devs.