(Edited 100% based on new photo-for previous answer see edit history)
Grounding here is a bit of a mess at the moment.
While that loosely goosey twisted together mess might work as grounding it isn’t good permanently and would not pass inspection.—The ground splice must be secured with a listed wirenut. (Over time in a closed up box expansion/contraction and oxidation can change how well things work with the “as-is” splice.)
The bare copper wire coming into the box with the black and white wire is the house ground. It needs to be connected to everything. Everything consists of...
- The silver fixture braid.
- The copper braid (I can’t tell where it comes from).
- The green ground screw on the yoke (pictured in pic#1)
- The box itself (currently using a ground clip.
How to fix:
First obtain a green ground wire nut, and some bare copper ground wire (pull a piece out of a scrap of NM-B wire.
Next turn off the power, and unconnected it all.
Using both braids and 2 extra pigtails cut from the bare ground wire and the house ground wire, twist it all together and green wirenut it. (You likely don’t have enough house ground to extend it thru the hole in the green wirenut, so in this example we won’t use that). This means 5 ground wires will be contained in the wirenut.
Now we have 2 pigtails coming from the splice, connect one to the green screw in pic 1, connect the other to the metal ceiling box—you have two options: either use the existing grounding clip, or even better if the box has a green screw or a place for one then use that. (Clips were invented for speed install, they take 2 seconds, the screw takes 5 seconds but is better.)
Always wrap wires tightly around the ground screw clockwise and tighten. Tight on all of these connections is important, over time simple twisted connections may oxidize and be less conductive.
The green ground screw in pic 1 that you questioned is not meant to be a point to connect fixture ground. It is meant as means to connect ground to the yoke and canopy. The purpose of this screw is often mid-construed, and has been as shown now—it’s NOT to supply ground to the fixture, instead it allows you to connect ground to the yoke and canopy.
It was once considered acceptable to allow the screws connecting the box to the yoke to supply the ground. That is no longer acceptable unless the yoke has a retainer/grounding clamp to better connect the box screws and is label self-grounding, yours is not.