What counts as "Thorns" damage for the barbarian "Tough as Nails" passive ability?

Solution 1:

Thorns is short for "damage done to attackers when hit by melee attacks", so you're correct. Any equipment or skill* that deals damage to melee attackers counts as Thorns.

The text you want to look for is "Melee attackers take XXX damage per hit."

This shield for example has 26 "Thorns":
Melee attackers take 26 damage per hit.

You can see your effective Thorns under the Details tab in your inventory.

Thorns: 48 damage done to attackers when hit by melee attacks.

*I'm unaware of a Barbarian skill that counts as thorns.