If my opponent moves into an uncontested battlefield and I play Rengar as a reaction, am I the defending player?

Yes, you are the defending player.

Your opponent is the Attacker — their unit moved first to the open battlefield, causing it to become Contested (Rules 445–446, and per 459.2.b.1 the player whose unit applied Contested is the Attacker). When you play Rengar, Trophy Hunter into that already-Contested battlefield via his special Ambush ability (which lets him enter battlefields with enemy units even if you have none there), you become the Defender (459.2.b.2).

Note this only works with Rengar, Trophy Hunter specifically. Rengar, Pouncing requires you to be the attacker ("I can be played to a battlefield you're attacking"), so he can't be played this way.

The full sequence: opponent moves → battlefield becomes Contested → non-combat showdown opens → you play Rengar, Trophy Hunter as a [Reaction] (since Ambush grants Reaction timing) → you're designated Defender → if both players still have units when the chain clears, the showdown transitions into combat with you as Defender.