If I understand correctly, the plate is hinged at the yellow line. (Equivalent to this, it could be pinned at any to points along the top edge.) This means that the edge is constrained along this line. No point along this edge can move. Consequently the plate cannot turn in its own plane, it can only turn around the hinge in a perpendicular plane.
The force in digram I is either wholly or partly perpendicular to the plane of the plate. (It could be a skew force, with components parallel and perpendicular to the plane of the plate.) The perpendicular component of force provides a torque about the hinge, which exceeds the torque due to the weight of the plate, and turns the plate.
In diagram II there is no component of force perpendicular to the plate, so there is no torque about the hinge. The applied force $F$ exerts a clockwise torque about the right-hand end of the hinge (black dot), but the weight of the plate, and reaction forces at other points along the hinge, create anti-clockwise torques which balance the torque due to $F$.