Here is a reference that may be relevant (you may know it already): Auslander-Rim has a paper called "Ramification index and multiplicity" . Even though they mostly discussed normal rings, their definition of tame ramification used only Hilbert-Samuel multiplicity, so it can be adapted to non-normal situations.

Here is a reference that may be relevant (you may know it already): Auslander-Rim has a paper called "Ramification index and multiplicity" . Even though they mostly discussed normal rings, their definition of tame ramification used only Hilbert-Samuel multiplicity, so it can be adapted to non-normal situations.