What additional considerations do pilots have when deviating from the planned routing on flights dispatched under Method 2?
As a Former Dispatch Examiner and an Airline Part 121 pilot for most of my career I think what you are looking for is this:

Unless the terrain is such that they must adhere to or close to the desired routing and they are notified of this, most deviations are inconsequential. It also depends on how many engines are on the aircraft. A twin engine aircraft must land at the nearest suitable airport in the point of time after an engine failure. Although it is wise to land with a greater than 2 engine aircraft, that is not the requirement. If an engine fails after takeoff before reaching cruise altitude the PIC should consider terrain clearance in his decision to continue, deviate or return. The PIC is responsible for his decisions and can deviate or not deviate if he feels his decision is as safe as the alternative. After drift down, most jets should clear all terrain with few exceptions but he should always consider obstacle clearance.

Remember everything in preflight must meet regulatory criteria. Once airborne the real world situation can alter and often does alter planning and some regulatory requirements. A lot of planning regulations go away after takeoff. The PIC is the final authority in all decisions after departure even though you both are considered responsible and should agree. If the crew has time to coordinate with Dispatch, then it is the dispatcher's job to warn the crew of all known hazards they may have overlooked while dealing with the in-flight problem even if they were briefed prior to takeoff.