"The First Amendment protects against governmental limitations on expression."
No, it doesn't. The First Amendment declares that the right of free speech, assembly, religion, and the press are reserved for the people, as opposed to the federal or state governments.
This has the very important effect of ensuring that the government cannot suppress those rights, but that's not all it does.
The fact that this is an established right means that public places cannot bar you based on your speech or prevent you from speaking because of what you say. For example, people going to public universities and pissing off snowflakes by holding a sign espousing religion or denouncing abortion or whatever else.
The degree to which a private entity is considered "public" (serving the public, open to the public, generally accessible to the public) determines what they can and can't do to people.
A few decades ago we decided that certain things need to be explicitly forbidden in most cases, such as refusing to serve people based on skin color, sex, religion, age, etc., refusing to hire them, refusing to consider them for housing, etc. In certain states, those protected classes extend to things like political views. The idea is that these things form a person's identity and they must be actively protected from those who would seek to deny them basic services and dignities.
If we're going to say a cake shop must make and sell cakes for same sex weddings (which is one step removed from serving same sex couples in general), then it's a no brainer to say that public forums, which Twitter, Facebook, etc. absolutely are, cannot discriminate based on those protected classes.
They can ban and censor based on what you say or do, specifically, but such enforcement must not be discriminatory. It can't even be fair but lopsided in who it affects, For an example, see the laws that affect specific groups disproportionately despite being written fairly, such as many attempted voter ID laws, the rules banning blood donations from "men who have sex with men", etc.
We recently completed step 1, getting these platforms recognized as the public forums they obviously are. Step 2 is legislation that explicitly calls them out and says they can't discriminate based on political identity, or a few court cases with the same result. This is what all the hullabaloo about "IBOR" is, and why "Diamond and SIlk" have been getting attention. Step 3 would be the fine tuning that falls into place as more cases are heard. For something like Twitter, it would revolve around "You banned user X for saying this, but didn't ban user Y or Z for saying this. Why?", or people accusing their algorithms of being biased, or people accusing their human actors of manipulating things or applying policy in a discriminatory fashion. We've already got a taste of that with the flak certain platforms have taken over their manipulation of news feeds and the way they push fake news.
An example using the second amendment would be that a private entity that is open to the public can tell people they can't bring in guns. They can even say certain guns are allowed and certain guns aren't. Maybe they only allow long guns and not pistols. They can't do something like whites can't bring guns but Asians can.
In short, a private entity, even one open to the public, can restrict many rights. But they can't discriminate along the lines of a protected class, nor can they apply a restrictive policy that is fair on paper but disproportionately affects certain classes.
[1] Twitter, Facebook, etc. are public forums. You are here.
[2] Political views / identity will be considered a protected class. Some places are sort of here. For example, California says you can't discriminate in employment due to political leanings.
[3] Fine tuning the details and setting specific examples/guidelines through inevitable lawsuits.