2016-07-13 : If the equipped creature dealt combat damage only to a player or planeswalker, Thirsting Axe’s triggered ability triggers. The same is true if the creature didn’t deal combat damage at all.

2016-07-13 : Thirsting Axe’s triggered ability checks whether the creature it’s attached to at the moment your end step begins dealt combat damage to a creature earlier in the turn. It doesn’t matter whether Thirsting Axe was attached to it (or to any creature at all) as combat damage was dealt.