If an opponent conquers a battlefield on their turn, then on my turn I charm their unit to my battlefield, and they respond with Ride the Wind to move it back to their original battlefield, do they conquer and score that battlefield again?

Yes, they do conquer and score that battlefield again. When their unit moves back via Ride the Wind, the battlefield becomes Contested, opening a new Showdown. Upon resolution, they establish control and conquer the battlefield. Since a rule states "once per Battlefield per turn" (meaning once per player's individual turn), they can score the same battlefield twice in one game turn—once during their turn and once during your turn.