Sure, but we're not in a poli-sci classroom and we're not (most of us) experts in it. Just as it would be rude to charge into a maths classroom and tell the professor that ackshually group can mean many things, it would be equally rude for her to keep loudly announcing during a party that people should stop referring to the "groups" for the party games since ackshually group means something formal and specific, right?
Not right. Since this is a political and highly politicised discussion about the Head of State and not a friendly get-together for something completely unrelated, I would expect people to take it seriously and use the approriate words. Nothing prevents non-experts from googling the terms in question and finding out what they mean in the appropriate contexts, it's not some hidden arcane knowledge.
Alas, we are a nation of amateurs in this as in so many other things. I did a pretty poor job of laying out my newspaper during those years when compared to anyone even slightly competent, for example.

I think everyone should be able to say how they feel about this stuff.
Plus, let's not forget: when someone says that this feels more like a presidency, they're not
wrong, they just mean that it has traits that they don't associate with monarchies. They're trying to tell you something about the change and why it doesn't appeal to them. Stop telling them they're wrong! Language is about communication, and they're communicating.
This isnt actually about understanding though, is it? When you and S:reu Briga call the post-HC King a President and post-HC Talossa a Republic, its not to further understanding. Its because the terms "President" and "Republic" are poisoned in Talossan parlance, not unlike the term "King" during Roman times.
I'm not sure "poisoned" is correct, but as I said, I think that people who like the proposed office would strongly prefer the label of "king" because it suggests that the change is less dramatic and because it might not scare off some people who generally prefer monarchy. I think that people who don't like the proposed office would prefer to call it a "presidency" because it more closely aligns with general expectations about that label and because it highlights how significant the change will really be.
I have to confess that I don't know much about German presidential elections, but Wikipedia suggests that they usually are partisan. [...]
I believe the quote you shared contradicts your assertion at least in part. And it looks like this long standing adage which I've never heard of, "if you can create a President, you can form a government.", is about to be disproven this year. Exciting stuff!
Again, I don't know much about the subject. If you assure me that German presidential elections have nothing much to do with partisan politics, I am inclined to believe you. The article is very clear that parties endorse candidates, campaign for them, and that the results not only usually correlate with the chancellorship, but are even seen as predictive... but it's Wikipedia! So I'd tend to believe you over the article, if you were sure.