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The subject doesn't lack rigor, but all of the popular and quite a few of the mathematically minded introductions to it do. Bourbaki and Chevalley did a good job introducing Clifford algebras properly some 80 years ago, but even this has been forgotten by the modern bloggy expositors.

This here seems to be one of the few good modern texts that are elementary yet rigorous: https://www.mathematik.uni-muenchen.de/~lundholm/clifford.pd...

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Unfortunately, the Lundholm and Svensson text you've cited suffers from the same problems I wrote about in my post. Definition 2.7 connects the interior products to the scalar product, not the inner product. Definition 2.8 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras.

It fixes the worst problem, which is the well-definedness of the interior product, since it defines the product explicitly on the standard basis rather than through a bunch of not-obviously-consistent axioms. It is too basis-dependent for my tastes (it should depend on the symmetric bilinear form, not on the basis). As for the sign conventions, I think they're ultimately matters of taste, and while I agree with you about the inner/scalar product, I disagree about the derivation-ness of a contraction (I like it to satisfy the Koszul sign rule, which allows signs to be inserted only when some factors move past each other).

My ideal approach is along the lines of Chevalley's "The Algebraic Theory of Spinors and Clifford Algebras", Chapter III, but he doesn't get to much geometric algebra.

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