If the NY times publishes an article that say AZjeff has done something bad (and you didn't) then you can sue them in court for damages to your reputation. Its called slander and its quite well defined. Free speech referenced in the constitution is political speech. You can't publicly say anything you want about anyone at anytime. The Sandy Hook parents were just awarded billions because a radio host kept saying the sandy hook killings didn't occur. He tried hiding behind free speech but the court ruled otherwise. You can't yell fire in a crowded theater, and you can't spread lies with impunity.
Twitter and other internet forums are protected against lawsuits because there considered the "public forum" in which these discussions take place. So while you could sue someone for what they post on Twitter - you can't sue twitter - because there the public square so to speak.
However when they start moderating or suppressing the information in the square based on their own beliefs they have now become the arbiters of the information. They don't simply say you can't talk politics - they allow political speech, but suppress some stories and highlight others. Therefore there now no longer the public square, there journalists. And journalists can and are often sued.
That's the difference many don't understand.
If they want to be journalists that is 100% there right, but then they shouldn't be protected by the special laws that protect public forums. Congress agrees actually, and both parties have been talking about modifying these laws - there woefully out of date.