As I described in my paper, the sum of the interior and exterior product is valid when one of the arguments is a vector field. It turns out with vector fields this is equivalent to the Dirac operator (it’s a theorem):
Theorem: \Delta^\frac12\omega = \mp\omega\ominus\nabla = \pm\nabla\wedge\omega \mp \omega\llcorner\nabla = \pm d\omega\mp\partial\omega.
This is the so called square root of the Laplacian, i.e. Dirac operator. However, in general this is not enough to fully understand what the geometric product operation is.
It is important to understand that the Dirac operator is an instance of the geometric product, but it is not the most general definition of the geometric product. This is best understood in terms of symmetric difference
Definition: \omega_X\ominus \eta_Y = (-1)^{\Pi(\Lambda X,\Lambda Y)}\det g_{\Lambda(X\cap Y)} \bigotimes_{k\in \Lambda(X\ominus Y)} w_{i_k}
The geometric algebraic product is the \Pi oriented symmetric difference operator \ominus (weighted by the bilinear form g) and multi-set sum \oplus applied to multilinear tensor products \otimes in a single operation.
As defined in this way, the geometric algebraic product is the foundational building block from which the other products are constructed. This is all also mentioned in my paper and the presentation.
The way I see it, the geometric algebraic product is at the root of everything. From this, the Grassmann and also the Leibniz algebras can be derived, on top of which you can also do Clifford-Dirac-Hestenes.
The sum of the differential and codifferential (Dirac-Clifford operator) is not the full definition of the product, rather it is an instance (or theorem) of geometric product when an argument is a vector field. The full definition is best understood in terms of the symmetric difference index algebra.
If my construction from the paper is used, then the Dirac-Cliford product is a theorem based on the underlying definition of the geometric algebraic product, which is more general.