This is not actually a First Amendment issue; as you correctly state, the First Amendment only limits government actions, not those of private citizens or organizations.
However, companies like Twitter currently enjoy some liability protections that are commonly understood to be based on Section 230 of the Communications Decency Act, that require them to act as free conduits of information posted by their users. If they exercise editorial control over the content, they may lose these protections.
Section 230 says:
No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.
The distinction between information content provider and interactive computer service is important. If the service provider exercises editorial control, they're generally understood to be the former. As I understand it, the distinction is intended to be analogous to the difference between a publishing company and a printing service. For example, a newspaper publisher can be sued for libel, but the company they contract with to print the newspaper would not be considered a party in the lawsuit.
While the First Amendment does not directly apply to the private sector, it does express the importance that our society places in free and open discourse. While there may not be any laws that prohibit Twitter from censoring its content, such actions may impact its effectiveness as a platform for this discussion. So even though they're not actually doing anything illegal, they're violating the spirit of Free Speech.
The Twitter Terms of Service says:
We reserve the right to remove Content that violates the User Agreement, including for example, copyright or trademark violations or other intellectual property misappropriation, impersonation, unlawful conduct, or harassment.
Notice that it says nothing about labeling content based on fact checking or political views.