Removal of stones in bunkers

Is there also no penalty if your ball moves if you move something like a loose twig when your ball is not in a hazard?

If you move a loose impediment (twig, leaf etc) and cause your ball to move and your ball is on a putting green then there is no penalty. If your ball is anywhere else, then yes, there's a one-stroke penalty. In both circumstances the ball must be replaced.

Note that the putting green exception is NOT part of the recently permitted local rule for balls moved accidentally on the putting green. This exception is a rule of golf.
 
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