That's a standard plasterboard backing box.
They have a thin flange around the front of them, which lands on the front face of the plasterboard.
Then, there are a pair of external sliding wings either side, the yellow bits in your photo.
You can't see the wings, they are on the outside of the box.
As you tighten the screws, the yellow bit will be pulled forwards till the wings clamp onto the back of the plasterboard.
The plasterboard is clamped between the front flange, and the wings.
The ole in the pb needs to be a close fit to the main body of the box, so the flange doesn't fall through from the front, and the wings can grab it from behind(!).
In order to get the box into the 'ole in the first place, the wings can be retracted into the box.
Usually, you slide the yellow bit all the way back, and it will then fold inwards.
The box is shoved into the ole, the wings pushed back out and the yellow bits pulled forwards till it touches the inside of the plasterboard.
Once the yellow bit is slid forward, it can no longer fold inwards.
Note that the final fore-aft position of that yellow piece depends entirely on the thickness of the pb and whatever other stuff may be stuck onto it.
Typically, there's nothing other than friction holding that yellow bit forward before the screws are tightened, and it will slide back quite readily.
So when trying to get the screw started, best not to shove too hard, or the threaded yellow bit will just shove away from you.
If it has, then just manually shove it back.
The wood screw you mention sounds like a bodge to stop it sliding away from you!
That it has a design feature (snapped) to assist with this sounds like an extravagance!
One thing about your photos...
The front flange is not visible. Is that some kind of facing which has been placed over the top of the plasterboard, and the flange is on top of the plasterboard, but behind the facing?
Never mind, I see you answered that earlier.
But that's why this needs longer screws.
