Ground wire from the junction box
The green screw firmly grounds the lamp to the mounting bracket.  You are asking  "How does the mounting bracket ground itself to the steel junction box?  Does only metal-metal contact suffice, or does it need to be wired also?" 

Yes, metal-metal contact between parts is acceptable grounding if all these are true: 


the mounting bracket is bottomed out "hard" against the metal junction box (not dangling by the screws or floating on drywall ears, as is usually the case with receptacles and switches)
and both are clean bare metal (not contaminated with paint, rust etc.)
and there are no insulators in the way (those little squares that outlets use to capture screws).


Unless there are any of those little screw-capture squares that I can't see, both your old bracket and your new one look like valid grounding paths to me.