Is The Federalist Society a conspiracy? They have open meetings. How about labor unions?
The dictionary definition disagrees. It adds qualifiers like 'secret' and 'unlawful' and/or 'harmful'.
to act in harmony toward a common end
https://www.merriam-webster.com/dictionary/conspire
That said, I don't think there's much question that the more common-place, colloquial definition includes the "secret" and probably "unlawful" aspects. But somebody who uses these terms in the other sense isn't necessarily wrong.
a similar example of a word that has become so loaded by the negative sense would be 'propaganda' - technically all political speech is propaganda, but in colloquial usage the term is so bogged down to mean roughly 'blatant and overt totalitarian propaganda' that we can hardly see the inherent political bias in 'acceptable' political statements, and are more prone to be influenced by them as a result
The actions against the black panthers, for instance, were secret and harmful to the group and the question of the legal status of those actions is open. The same could be said with regards the US treatment of the American Indian Movement, or many, many actions taken against labor organizers in the late 19th C.
Much could be said of about anything the CIA has done in its history.
IMO, it's more useful to just open up the term to any organization that acts in concert, because then a lot of the fuzzy distinctions between, say a government that isn't acting in the direct interests of its entire population and a criminal cartel.
Those are clearly different kinds of organizations, but "harmful" and "legal" become questionable in many situations.