Listing inner-department seminars/presentations that focused on presenting current research by other authors? At my university there are weekly seminars where faculty or students can do presentations for the department.

I had done several of these during my time as a graduate student, although admittedly they were seldomly composed of original work.  They were often taking a recently published paper and presenting the findings, sketching the proofs, and illustrating how it applies to research you are interested in and/or doing.

I started to list these on my CV as a more generic "Department Seminar Presentations" of a given paper/topic, but as I thought about it more I began to wonder if this is actually improper to do?

This is in mathematics.
I don't think it's improper per se, but I would not include them.

These sorts of presentations are more akin to assignments for a class: they can improve your skills in terms of public speaking and also getting deeper into the material.

However, you wouldn't put your course assignments on a CV, and I wouldn't put these presentations on a CV either.

I wouldn't even include them if they were based on original work, unless they are in a format that indicates some level of acceptance or prestige: conference presentations, invited talks at other institutions, etc. Instead, you reference that original work where it appears in public: in publications, for example.

They might be appropriate in a more generic form in a non-academic CV, when the point is to emphasize your experience presenting these sorts of things to a public audience, rather than your academic credentials.