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.
agree.. fragmenting words by colloquialism tends to obscure the deeper and more complete 'sense' that existed before the fragmentation - so we're left with more precise words, but less linguistic cohesion
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
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.