For Verity Circle's first ability to trigger, the creature has to actually change from untapped to tapped. If an effect attempts to tap a tapped creature, but it was already tapped at the time, the ability won't trigger.

If a creature enters the battlefield tapped, Verity Circle's first ability doesn't trigger.

If a creature with vigilance is declared as an attacker and then becomes tapped while it's attacking, Verity Circle's first ability triggers.