The word director is used as in the director of a film as this is a good analogy of what the role entails. This role is somewhat similar to the Product Owner role in Scrum. Your responsibility as Product Director (PD) is to communicate and maintain: -
a product vision
a wall of work containing high-level work items
The FAST Product Director is a key leadership role. Developers will want to follow their vision and build their product. The FAST Product Director should be able to "rouse the troops", so to speak, at each iteration boundary.
The tribe is made up of cross-functional skills and T-shaped people. Your role name is "Developer" regardless of what your primary skill is. (This is the same as Scrum.) You are expected and encouraged to be excellent in your craft. If you are writing code, you are expected to be a Code Crafter and quality code is expected. You are also expected to be a coach and be coach-able. No holding on to knowledge or trying to creating silos. You should always be working on both aspects of your T-shape - generality, and specialty.