This is a modern reappropriation of the phrase, really. There are two issues: what should the government be allowed to do to limit speech (historically, this was called "free speech") versus how companies should be compelled to police or host objectionable content (this is the "new" connotation).

It's unfortunate that these two concepts are often lumped together in online discussions, because they are obviously very different, and many people who would agree with the First Amendment and the classical notion of "free speech" as a restriction on the government could have diverse opinions on the regulation of platforms and how they display content.