The definition you linked to says nothing about 3rd parties, only officials without specifying who they work for. In fact, one of the examples given in that definition is film censors which is within scope here - in the US film censorship is done by the MPAA which is an organization composed of movie studios. Most of the films they censor are produced by their own members.
The use of the word censor in this discussion is entirely within the common and technical definitions of the term.
http://variety.com/2013/film/news/new-york-theater-will-admi...
Though most theaters do abide by the ratings, and therefore movies will continue to modify their content until they get at least an 'R' rating. https://en.wikipedia.org/wiki/Motion_Picture_Association_of_...
The MPAA issues ratings; it's up to the studio to decide whether to make cuts to a film in order to attain a particular rating. No-one forces the studio to make the cuts (Wikipedia: "The MPAA rating system is a voluntary scheme not enforced by law and films can be exhibited without a rating").
Sure, it might be commercial suicide to issue a film with a particular rating, but the right to free speech != the right to make lots of money.