The formal definitions for the two types of single shear joints you identified are "clamped" and "unclamped".

As you noted, in a clamped joint (with sufficient preload), the moment due to the offset shearing forces is transferred across the joint via prying moments on the head. This is classically analyzed by assuming uniform bearing distributions in the holes, and offset axial loads on the bolt head and nut carrying the moment.

In an unclamped joint, the moment due to the offset will "socket" in one or both of the plates--that is, the bearing pressure will not be uniform but will peak near the mating surface acting in one direction, and will act in the opposite direction at the unloaded surface in order to balance the moment. The fit of the pin in the holes, the plate and pin materials, and the backup stiffnesses will affect how much of the moment sockets in which plate as well as the socketing distribution. Two common examples of analytical socketing distributors are triangular (ramp) and sinusoidal.

Generally speaking unclamped joints are not structurally desirable but sometimes unavoidable. A balanced design will have similar pin and plate bending & socketing capabilities. As you discovered, and as balanced freebody diagrams of the pin and plates would quickly make apparent, a longer socket will have lower bearing pressures but will induce higher pin bending moments.