When I move an enemy unit to a battlefield I control, does that make the enemy unit an attacker?

Yes, the enemy unit becomes the attacker.

According to RiftJudge FAQ #5394 and FAQ #7275, when you move an enemy unit into a battlefield you control, that unit is treated as the attacker. The enemy unit is considered to have moved to contest the battlefield you control, and therefore it applies the "Contested" status to that location, which designates it as the attacker regardless of what card or ability caused the move to happen.

This is distinct from moving a unit to an empty (uncontrolled/unoccupied) battlefield, which initiates a "Showdown" rather than "Combat," and does not result in an attacker/defender designation (as detailed in FAQ #9883 and Rule 429.1).