Harsh Mentor
Whenever an opponent activates an ability of an artifact, creature, or land on the battlefield, if it isn't a mana ability, this creature deals 2 damage to that player.
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Official rulings (3) · Scryfall
Harsh Mentor's ability resolves before the ability that caused it to trigger.
Harsh Mentor's ability doesn't trigger when an opponent activates an ability of a card in hand (such as cycling) or a card in a graveyard (such as embalm), even if that causes a card to be put onto the battlefield.
Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keyword abilities (such as equip) are activated abilities and will have colons in their reminder text. An activated mana ability is one that produces mana as it resolves, not one that costs mana to activate. Notably, exerting a creature isn't an activated ability.