This ties into the part you absolutely agreed with. The word "God" as it is defined now would not exist without the original unproven claims that God. Even if you're not responding "God doesn't exist" directly to someone who said "God exists", you are if nothing else still responding to the original millennia old claim that they do exist. For that reason, it is always a counter-claim.

If I say god doesn't exist to you I feel like I'm making a true or false factual claim to YOU rather than to a bunch of old dead people or some greater historical/cultural context. The history of the word/definition might be relevant for deciding what the word means, but the claim is aimed at YOU. The actual truth status of the claim (even if we call it a counter-claim) that I might be making is either true or false (assuming we subscribe to bivalence like that) regardless of the history or culture that lead us to the discussion.

As for what makes counter-claims different from regular claims, it's simply that the burden of proof lies first with the original claim. A counter-claim has no responsibility to prove their claim until such time as the original claim presents evidence supporting itself

It seems like a silly double standard for only one side to have a burden to prove their claim, but the other gets to claim the negation is true with no burden of proof.

For example, if you say "2+2 is 4" and my response is "NO IT IS NOT. IT IS 3! I REFUSE TO PROVE IT THOUGH", not only will I be wrong in a classical arithmetic sense but I have presented no argument for why you ought to believe my new counter claim to your original claim. It would make no sense to believe me without more info in such a case.

The problem with that is I at least in theory could have looked up the tax code, remembered it, and then told you it correctly. Sure, I could have lied or remembered wrong, but it was 100% within my capacity to give you the accurate information, and even show you where I got the information from. With a claim about God's existence, that's impossible for either side of the debate as far as we know, and since the original claim was "God exists", that side is, possibly forever, stuck holding the burden of proof.

The fact that you can look up tax code is not really a problem for my hypothetical example. It is not particularly hard to come up with hypotheticals where you just can't easily obtain the answer. We could rephrase the context, perhaps we are stranded on a desert island? We could rephrase the question, perhaps it is about what some obscure historical figure had in their pockets on the day they died?

To be clear, I'm not trying to argue for or against the existence of god. My issue is that there should be a burden of proof for the CLAIMS "god exists" and "god does not exist" if somebody is claiming either is true. I don't think there's any kind of burden for believing some random claim without proof, but I think it's silly to commit to the negation of a claim without proof unless you have a reason to believe the negation. You can always just not commit and say you don't know in such a case, rather than believing the claim or its negation.