What do the white dumb-bell signals indicate?
I keep thinking of some of the big fly ins (like Air Venture in Oshkosh) where pilots are allowed to park on the grass.  

During the fly in they might use that second symbol (dumbbell with stripes) to let people know it's okay to park on the grass, but you still need to take off, land, and taxi on official taxiways and runways (as defined by the event rules).  Thus significantly increasing capacity during the event.

But during other weeks that same airport probably doesn't want anyone taxing onto, or parking on, the grass.  So they would display the first symbol (dumbbell with no stripes) to remind everyone that the exception is gone, and the rules have changed back to what you would normally expect at most airports.  Thus people would be aware they need to stay off the grass and use official tie down areas for parking.

In essence, I think the first symbol exists mostly because of the second one.  At airports where there are occasionally exceptions along the lines of those described above, they may also want a symbol to help pilots know when the exceptions are not in effect.