This is an important question that affects consumer advocacy, blog-based-journalism, political speech and spending, and probably other things.
People disagree about what the answer is.
This podcast episode features both sides of the argument.
Professor Volokh holds the view that the freedom of the press is "freedom to print", "freedom to use the printing press". Freedom of the press is the freedom of everybody to use the printing press. Assessing the grammar of the clause, he argues that this points in the same direction. The text says "freedom of speech or of the press". He points out that speech is not a group of people. It would be odd to treat "speech" as an activity, but to treat "press" as a group of people given the parallel construction. This side of the argument is described more in this article by Prof. Volokh.
Professor West argues that there is a defined group of people called "the press" that deserves protection under this clause. This article explains Prof. West's position in detail. The main point is that "An expansive definition of the press means virtually complete overlap between press and speech and thus no meaningful way to interpret the Press Clause." If "press" means simply the right to publish speech, then it becomes redundant because courts have held the right to publish speech is given under that "speech" portion of the clause. Justice Stevens's concurrence in Citizens United also argued for "some kinds of identity-based distinctions" regarding whether a person is a member of the press.
Each side can give examples of the term "the press" being used at the time of the First Amendment that is consistent with their favored interpretation.
I think to get the best idea of the two sides to this question, you should read Citizens United (including all dissents and concurrences), read the two articles linked above, and listen to the podcast episode.