Thirsting Axe — ruling 1

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.

Thirsting Axe — ruling 2

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.

Thirsting Axe : règles, fonctionnement et interactions (MTG)

Thirsting Axe is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

Equipped creature gets +4/+0. At the beginning of your end step, if equipped creature didn't deal combat damage to a creature this turn, sacrifice it. Equip

How it works : Thirsting Axe

Original English text — Wizards of the Coast / Scryfall

#1July 13, 2016· Wizards of the Coast
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.
No other card is named in this ruling.
#2July 13, 2016· Wizards of the Coast
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.
No other card is named in this ruling.