All three are different laws, yes, but the latter two also make it clear that service providers are immune from liability for third party content in those contexts if they aren't actively participating in its selection or alteration, excepting justified deletions.

You said it's completely false that exercising control over content on one's network diminishes a provider's ability to claim safe harbor. I've given you two laws on the books in the US that say otherwise. What are you basing your claims on?