Based off your response, I decided to try re-training my remote, being careful to hold the button on the apple remote down for as little time as possible (so the remote wouldn't have a chance to register more than one key-press). After doing this, and being extremely careful to barely tap the button, the problem actually got WORSE.

This gave me an idea, so I went back and re-did the IR learning, but instead of holding the button down as briefly as possible, I went the opposite direction, and held each button-press for a quarter second. Et viola, it worked! Now the remote works exactly as it should.