The connotation is a product of how the label was inconsistently applied in the past. Consistent application of the label will solve this problem pretty soon, without requiring Twitter to make judgment calls that should be beyond its purview

A connotation is subjective and should not matter. Public funding however is. In my country there are more or less trustworthy government funded news stations that are also flagged. At least on youtube.

But I believe the label is more important than what some people fear other people would maybe perceive.

It would simply be a connotation if Twitter's own documentation on what that tag means didn't explicitly state that the tag means that the source's editorial process may be influenced by the government organizations funding them. That's not the case for any of the current organizations that have gotten that tag.

Look at the difference between how @BBC is labeled and how @NPR is labeled and try to reason why they're not the same.