If I control a battlefield, move my units out, then move a unit back in, does that count as a new conquer?

Ruling: No, it does not count as a new conquer. Conquer only occurs when a player gains control of a battlefield they did not yet score this turn.

Nuances:

• Once you have scored a battlefield in a turn, moving units out and back in does not trigger another conquer for that same battlefield in the same turn.