I don’t like the idea of calling people or organizations “press.” The first amendment says the government can make no law restricting the exercise of the press. It did not say the government can’t restrict “the press” as if it’s a body of people. “Pressing” is the act of publishing and distributing information and ideas. All people have the right to do so under the first amendment.
I hear journalists say sometimes that their profession is the only one specifically protected by the Constitution. They then use that to say how important they are. That’s an arrogant misuse and misinterpretation.
People and organizations may call themselves press. But that very obviously wasn’t the definition used at the time of the Constitution. And journalists are not a special class of citizens. As much as many of them would like to believe.