Something's not right here. From your other photo with the card inserted, it looks like a single B-key socket. The  M-key doesn't seem visible.

I'm also puzzled by any card width difference between one side or the other from your other photo of the non-chipface side of the card. The slot may not line up with the key, but the card width should be the same on both sides.

You also mentioned the socket pin contacts were on the bottom. I guess I'm not really seeing that. Are you first fully inserting the card  at about a 30* angle, pushing the leading edge toward the bottom of the socket, then pushing the screw end of the card down so the card contacts push up against the socket pins? The pins should springy

 

OK. How about the 30* angle of initial insertion into the socket while applying slight downward pressure on the card's leading edge.

I fear this socket is beginning to look more and more like a B-key SATA socket. Unfortunately, this might mean an NVME card might not physically fit because it has no B-slot that aligns with the B-key.