I can't figure out what the wire labelled "functional earth" is for.
All earths should be 'functional' & all should be bonded together.
TL:DR - No, don't do it like that.
It is not sufficient to use the patress screws as your bond [which is what you appear to be doing in your photograph]. The ring should be physically connected to earth at both ends, but this is potentially removing the continuity, which is unsafe. A ring has redundancy built-in, so don't intentionally remove that redundancy.
Your ring main earths should both go into the Earth terminal[s] on the socket. The diagram shows two earth terminals, bonded to each other & also to the patress screw-holes. This is good. In practise you can use one or both, but don't crowd one just for the sake of it. From either of those terminals, then run a separate 6" of green/yellow sheathed wire to the Earth tab on the back box.
Earth bonding in the socket…
All earths need to connect to this - mains in, mains out, back box [& whatever is considered a "functional earth"] This means the patress screws & back box can never accidentally go live, even with the socket removed from the back box.