This particular set of options doesn't reflect the immunity in §230 of the CDA, which both immunizes intermediaries and authorizes them to remove content they find objectionable. (While that doesn't mean that intermediaries can't be required to be neutral by other legislation or regulations, the particular either/or principle you describe hasn't been the law in the U.S. for a couple of decades.)
47 USC §230(c)(1) provides that "[n]o 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" and then (c)(2) immediately says that "[n]o provider or user of an interactive computer service shall be held liable on account of [...] any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected [...]".
Edit: to be precise, the content immunity in §230 isn't a generic common-carrier immunity; if you read the introductory text, it's a very Internet-specific immunity for "interactive computer services" such as Internet services!
https://en.wikipedia.org/wiki/Section_230_of_the_Communicati...
47 USC §230(c)(1) provides that "[n]o 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" and then (c)(2) immediately says that "[n]o provider or user of an interactive computer service shall be held liable on account of [...] any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected [...]".
Edit: to be precise, the content immunity in §230 isn't a generic common-carrier immunity; if you read the introductory text, it's a very Internet-specific immunity for "interactive computer services" such as Internet services!