Homer Alone - S3-E15
Deliberate mistake: Arnie says that he drops his bagel. When he does he starts to say it, but when we see the helicopter it has just fallen out. If he dropped it at the time he did, then it would be much further down and wouldn't be visible in that shot. (00:05:20)
Deliberate mistake: Lou and Eddie are driving Bart around in the cop car. Bart asks when they're going to shoot people. The shot cuts to Lou, who is driving. The entire frame of the car disappears around him and it's just him superimposed over a background. The shot cuts again and the frame is now visible. Even if it was a low-angle shot, you'd still see the roof of the car.
Answer: Willy referred to an encounter he believed he had with actual aliens, while Wiggum thought he was referring to the arcade game.