What is the term for a gambrel roof with an extra angle on each side?
I still consider it a gambrel. The word simply means "leg" or "forked leg", describing the interior supporting timbers.

The inherent flaw in the design as you've shown it is potential lack of rigidity at the upper angles. Unless they're robustly gusseted there and at the lower angles they're in danger of inverting at those points. There isn't enough arch to prevent outward splay and maintain compression.