Ahhh, I see the problem it your socket needs the tip ground down a little. It is a problem that many sockets have a bevel on the leading inside edge which will not grip shallow head screws. This is why I have several small diameter sockets with the tips ground down.
Something I had forgot about as it is automatic thing for me to do from my years of working on office equipment.
BTW the 5/16 and 8mm socket are virtually the same size sockets as is 7/16-11mm, 1/2 -13mm, 5/8-16mm, 7/8-22mm, and 15/16-24mm as long you are using the six point versions. It when you go to the 12 point versions you will see the difference.