Do the counters that Ring of Thune adds stay with the ring or the creature?
The Ring of Thune puts counters on the creature, so the counters stay with the creature, not the equipment. If they wanted the power and toughness bonus to stay with the equipment instead of the creature, they would use a wording like on Sigil of Distinction:

Sigil of Distinction enters the battlefield with X charge counters on it.
Equipped creature gets +1/+1 for each charge counter on Sigil of Distinction.