The problem, in a nutshell: 
The Harmony Remote doesn't always correctly detect Apple Remote presses when programmed. If that happens, it'll just play back the received, raw IR codes multiple times instead of sending a single, correct Apple Remote code (like a real Apple Remote would do). This results in the Apple IR Receiver receiving the signals of multiple, quick Apple Remote button presses for what is actually a single press of a button on your Harmony Remote. And for the signals of multiple button presses, you get the response for multiple button presses.

Solution: 
Use the Apple Remote profile as supplied by Logitech. It's been correctly programmed and will send only a single, correct Apple Remote code per button press.

Best regards, 
Felix Schwarz