What you are talking about is "white propaganda": telling the truth but in a manner that creates a useful perception.
This is unfortunately the basis of most information and news media today, as even "free" media industries compete to tell the stories and gain the access of intelligence and national security reporting (how we got the Iraq War propaganda in the US, how today the media industry is spreading "Russia did everything" nonsense, the current media sprint repeating propaganda about the situation in North Korea and the DPRK government's intentions).
> So now the counter-propagandist is put in the difficult position of arguing against the publishing of the truth, not a particular nice place to be, rhetorically.
Exactly.
I would emphasize that "truth" is not a binary thing but a highly contextual and multidimensional device. Merely changing a headline or picture sets tone and dramatically alters perception - add to that editing, media campaigning, social media astroturfing, selective synonym choice...
"Perception management" is an regular propaganda term employed by the Department of Defense for this reason. The facts can be useful, but ultimately the objective is to change minds and decision making and to affect outcomes. And so you may be the subject of perception management that utilizes (aspects of) truth.
The most effective way of countering propaganda is to seek nuance when presented with characterizations, try to understand dispassionately the situation from every side, and to be skeptical of especially the things that you most routinely would like to agree.
Ironically this usually means consuming more propaganda on purpose: specifically looking for the narrative position of the United States, Russia, Britain, China, etc on various issues. This is also why it's an extremely good idea to read international news from all parties in a conflict during the entire duration of an event (e.g. Syrian Civil-Proxy War).