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I think my driver side exhaust valve rattles when cold but the dealer said the gap/play in the valve is normal (even though the passenger side valve do not have that play). See below pic. Anyway, it only rattle a minute or 2 before the car got warm then disappear so I am not to picky about it.
1st pic is the driver side valve with gap, 2nd pic is the same valve without play/gap
Yours look a bit different than mine, I'm not sure if there's a nut on top of the "axis" in the center of the tube, but if there is, you can try adjusting that, otherwise, you can easily adjust the length of the actuation arm by removing the small clip at the ball joint, popping the cup off the ball and then twisting to lengthen or shorten to add a bit more tension so the valve is always tight.. There's also a lock nut on the arm after you've adjusted it.
Yours look a bit different than mine, I'm not sure if there's a nut on top of the "axis" in the center of the tube, but if there is, you can try adjusting that, otherwise, you can easily adjust the length of the actuation arm by removing the small clip at the ball joint, popping the cup off the ball and then twisting to lengthen or shorten to add a bit more tension so the valve is always tight.. There's also a lock nut on the arm after you've adjusted it.
The pictures posted by Lowbee are from the OE back box and look exactly like mine. I don't have the left/right variation that he is showing, but mine rattle A LOT during cold starts and I can't wait to rid myself of that factory box.