Visible crew/equipment: When Spider-Man is rescuing Mary Jane from falling to her death from the balcony, he catches her and spins his web to the balcony to save her and they bungie to just inches from the ground. As they are preparing to bounce back up the shot rotates 90 degrees and you can see the counter weight that pulls them back up coming down to the ground. It's a weird one, this - there's a sound effect as the weight comes into view, so it's meant to be there (or else the sound's just a cover), but it can't be anything he attached his web to, as it was stuck straight onto the balcony. If it's falling, they should be too - his web's not pivoting around anything. (01:06:50)
Visible crew/equipment: When Peter stands up after being bitten by the spider, there's the reflection of the cameraman with headphones on the television set behind him. (00:10:00)
Dr Wilson
Visible crew/equipment: When Peter wakes up and discovers he can't see through his glasses, you can see a phantom hand on the left sliding out of view in the mirror. This happens the last time Peter tries to see through his glasses. It's not a reflection of his hand due to the angled edge - it's on the main part of the mirror. (00:18:05)
Visible crew/equipment: When Mary Jane is on the balcony at the Oscorp Festival and the Green Goblin causes part of it to start falling apart, the wires and chords causing the balcony to fall down and pull back up can be seen. (01:07:20)
Visible crew/equipment: When the fire breaks out in the building, as the firetruck goes by you can see the reflection of the camera crew on the truck, including a man wearing a white shirt. (01:22:00)
Visible crew/equipment: At Thanksgiving dinner, when Aunt May looks outside when she sees something, the reflection on her glasses shows what appears to be a green screen. (01:26:00)
Answer: The gesture isn't to create the web, it's to actually shoot them. As such, if another method of drawing them forth is available, the gesture isn't needed. The fork's managed to get stuck to the exit point for his webs - we see what appears to be a small patch of a weblike substance, so presumably it's slightly sticky. Peter then pulls the fork, and in doing so is pulling the webs out from the spinnerets - so no need for the gesture in this case. When he then webs the tray on the next table, if you look closely, he's doing the correct gesture.
Tailkinker