Not all pedantic arguments about the definition of words are bad faith. Sometimes, the original argument genuinely employs a confusion of definition, this could make the argument void, see section "Equivocation" under <http://enwp.org/List_of_fallacies#Informal_fallacies>. This confusion happens both unintentionally and intentionally.

I would like to propose to not treat the pedantry counter as a sin. Instead think of it as a weak signal: the counter does not rank high on the hierarchy of disagreement. <https://bigthink.com/paul-ratner/how-to-disagree-well-7-of-t...> If a strong counter could be made, we would use that one instead of the pedantry counter.

I'm not even sure who is being criticized here. I think it's just an issue of tone. Bluntly telling someone they're wrong, when their statement is arguably correct, will generally trigger an argument over who is correct.

To call a distinction pedantic is to say it lacks importance to you. You are seeing a clash of values, not a bad-faith argument.