How to put the washers on the screw
The "rings" you mention are commonly called "washers".

It's hard to tell from the photo, but I believe the "incomplete ring" you mention is a "split" or "spring" lock washer:



If that's the case, then the are stacked as you have them in your second picture, the flat washer against the bracket, then the lock washer on top of that.

Note that this assumes that you don't have a nut that goes on the screw, if that's the case, then it's possible that the lock washer goes against the nut and the flat washer goes under the bolt head.

Based on the diagram you provided in the edit, it looks like the diagram wants you to put them in this order:



That looks like it's a wooden bedframe, so the flat washer is there to give a bigger surface for the bolt head when it screws down onto the soft wood.