From a free speech/free press standpoint, private company Twitter absolutely has the right to editorialize Trump's tweets, while Trump trying to silence Twitter would be the government infringing on the right to free speech/press.
Exactly. It is just another lie that Twitter would be "stifling free speech". Free speech was not stifled: Trump could even say what he wanted even though it is a private platform.
When referring to free speech, it is common to refer to the moral backbone of freedom of speech (strong entity shouldn't be able to silence the masses) instead of just the legal/constitutional definition.