They re called both...
Depending on the harness ,there can be multiple rings. On the harness saddle if used ,imagine two wood poles attached to a horse one each side, attached to the harness saddle. (Nothing to do with a saddle for a rider.)
Terrets were also used on a collar. We see more old collars around than harness saddles.
Just to make it more interesting there were some decorative extra terrets added to some collars. (Maybe harness saddles too(?)) Swingers ,fly terrets ect.
Reins pass through rings on both ,if both are used together. Or just on whatever individual , either harness saddle or collar is being used.