There is simply no objective definition for either of these concepts and if you may be fine with Twitter having this kind of power now, but you may regret it if the shoe is ever on the other foot.
Having a corporate arbiter of truth is antithetical to a free society, especially when creating one's own platform is impractical because of network effects.
Who is going to fact check the fact checker? Effectively Twitter has the final word because this is their platform - not because they are correct, and if these political squabbles had an objectively correct answer then we wouldn't have multiple parties.
People are find with it right now because it's en Vogue to hate on the president, but that doesn't mean the fact checking is justified or even honest - and I reiterate, the dangerous part here is that people implicitly take Twitter to be an authority when they absolutely are not. Their fact check isn't even accurate, but people don't tend to question what they want to hear.
Twitter's role is a neutral party - the same way that journalists are ultimately doing a disservice to society by editorializing everything while claiming to be neutral. It is dishonest and divides society from the top down.