Free speech as an idea doesn't just apply to the government, it is just that we only enforce free speech on the government. The ideal of free speech and its merits applies just as well to private actors and we should try to live up to it whenever we can. If you think that companies should be able to use their power and influence to suppress the speech of individuals then I would say that you are more authoritarian than liberal.
I'm so baffled by this because there are plenty of places where speech is suppressed and its fine. A highschool teacher can fail a student for shouting answers in an exam. A person can kick someone out of their house for saying horrible things. A cafe can ask someone to leave for saying slurs. A professor can ask a student to leave a lecture hall for talking...
The right to speech is curtailed all the time in private... We even teach it to our kids, such as raising one's hand, or waiting their turn.
You are misunderstanding free speech, it isn't a literal statement. Free speech is about being allowed to express ideas, not disrupting classrooms or harassing others. If you are allowed to express your idea after waiting for your turn, then your speech wasn't suppressed even if people told you to be quiet for a while.
It is fine to police disruptions, but you shouldn't police ideas. If all conservatives are told to wait for their turns while it is fine for liberals to just blurt things out then it isn't free speech. However if everyone is forced to wait and a conservative gets banned for talking out of turn then it is still free speech, he got banned for disrupting and not for his message.
It is hard/impossible to create laws around it since it is hard to formally define, but often it is obvious when it is infringed in practice just that we can't litigate it.
Or maybe we think that giving the government the power to tell private companies which speech they must allow is more authoritarian than letting the companies decide for themselves. Whenever there are tricky balancing questions like this, I always err on the side of the party that doesn't have a monopoly on violence.
Extending free speech laws to cover companies which acts as public forums is not the same as the government telling companies what they are allowed to say and do. If you think that free speech laws are fine for governments then you should also think it is fine for certain companies which grew too powerful and influential.
How do you think those laws will be enforced, if not the government telling companies "you must allow this defamatory and/or untrue content on your servers"? Facebook and Twitter can't censor me, because I am not a customer of either company, and neither has a law enforcement wing. The government, on the other hand, has unlimited firepower and I have no choice whether to "do business" with it. So yes, I will absolutely apply different standards to each.
Why is it up to the company to decide what is defamatory or untrue? If you read the rebuttal twitter posted on his tweet you’ll see it boils down to “no evidence”. This doesn’t mean his claims are not true, it means there’s no evidence they are. These are very different concepts. So now twitter is running around saying it’s untrue when legally it hasn’t been proven untrue. They should instead take a hands off approach and let people think, read and decide for themselves.
I personally feel much freer to think for myself if government officials can't force private companies to carry their personal content. Of course it's highly unlikely that Twitter fact-checking the president will make any difference - everyone who's been paying attention made up their minds about the guy years ago - but if Trump is really so triggered by it, he is of course free to post his thoughts elsewhere. It's awfully telling that he immediately decided to involve federal regulators, although I suspect this will be just as effective as the fact-checking.