It should be a conspiracy of like-minded individuals that exists parasitically within Starfleet, not an official (or an “unofficial official” agency).
I agree. When 31 was first introduced, and Sloan explained that Section 31 was sanctioned by Starfleet under Article 14, Section 31 of the Starfleet Charter, the implication was that they were people who misinterpreted or construed a (probably minor) part of the Starfleet Charter and used it to justify damn near anything.
Personally, I hate how Section 31 has been changed to be misunderstood, cool good guy/anti-hero types who are doing the wrong things for the right reason. DS9 had it right with portraying them as the villains within who should be snuffed out because the ends don’t justify the means.