If an axle with wheels rolls such that the left wheel is on pavement and the right hits grass, it will turn right.
A glass on a bar table top doesn't melt the surface to create a film; more motion means more friction not less. So, opposite.
It probably doesn't matter which edge (front versus back) has more weight on it, but rather which side of that edge (left or right) experiences more friction. I suspect even if you slide the glass up an inclined bar table top so that the weight is on the rear, it will still curl opposite.