The instability issues will most likely never get solved because it is very hard to avoid that problem in a rigid object only physics simulation for reasons i won't go into here.
Even better is to observe the fork while it closes and as soon as you see it doesn't move anymore even though it might not visually hit the trunk boundary, just release it a little bit and lift off. Doesn't always work as sometimes depending on log weight and diameter it still slips off the fork and you have to pick it up again. Just release the fork a little bit at that moment and you should be able to lift and transport it fairly well.
That is the moment the collision physics gets wonky because of intersecting objects. Also while you close the fork to grab the log, try and notice as soon as the log starts to uncontrollably vibrate just a little.
I too found that it works fairly well if you approach the log with the fork from the top or at an angle of say 30-45 degree (angle between ground and lower part of fork).
