Not sure if it's 'normal,' but I think that lean and the extent to which you flare your elbows have much more to do with chest/triceps feel than grip. In my experience:
For Chest- Forward lean, elbows flared
For Triceps- Stay as vertical as possible, elbows tight to the sides
When I do use dips (sparingly these days), I tend to shoot for a triceps feel. I don't have any effective way to hang weight off of myself, so the load would be pretty weak for chest work.