The point is: there is no natural or distinguished isomorphism from a finite vector space to the vector space of linear functionals over it. We know they are isomorphic- but there are many such isomorphisms with little to distinguish between them. The note's "arbitrary multivector, G" fix for inner products is good. The idea is most of us are using the same G that looks a lot like an identity matrix. And as long as you always use the same G you uniquely identify the inner product by equality (as the note does). But if you accidentally combine work that is using different G you run into trouble. You can end up with different Gs if you don't worry enough about naming your basis.
A lot of this keeps getting re-invented as variations of tensors, the exterior algebra, wedge product and so on. I think most of these end up being more bookkeeping than consumers want.