Twitter is a private company. They can censor whatever the fuck they want in any fucking way they want. Don't like it? Start your own bible thumping koch sucking social network platform.
The problem is not private/public. It is that they are immune from libel laws because they are simply a platform. Once they start editing, they become a publisher... not a platform. Their immunity is gone.
The other platforms know this. The person who owns Twitter owns CNN. This should tell you all you need to know if you weren't insistent on blaming Trump for this.
They cannot fact-check Trump exclusively in the political arena and get away with it. It will cost them, in the end.
It reminds me of states that were opening businesses, but not allowing churches to open. The DOJ made it very clear how they cannot selectively apply rules to churches and they'd get bitch slapped if they did.