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.

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.