> Free speech is a about whether a society values and protects free speech or not.

That is a recursive and meaningless definition. Free speech is widely understood, and codified, as being able to say stuff without having the government send you to jail for it. That is a pretty basic requirement for a functional democracy. Altough, depending on where you live, even free speech does not mean fully free. In the U.S it is mostly limited by libel/defamation, but in a lot of country hate speech is not allowed. Whether you think it is a good thing or not that is up to you, but this is actual free speech.

Yes, the First Amendment is about the government.

Freedom of speech, in the other hand, is part of a moral code that believes in inalienable rights, that humans implicitly have the right to express themselves. The government does not grant the right to freedom of speech, because we already have it. The first amendment says that the government must respect that right, but creating the right.

> Free speech is a about whether a society values and protects free speech or not.

That's a bit vague. Nothing about this statement on its face would preclude some speech regulations so it reads more like an empty platitude to me.

Perhaps but it's because you two are talking past each other. The free speech some people refer to are that devised under the concept of classical liberalism which spawned the bill of rights, under which the right to free speech is a right to not have your speech interfered with by government. Not a right to compel others to display your speech, as under that system you would be violating their right to free speech by making them say that was actually what was just your speech.