It is a gray area and social media platforms sit somewhere in-between being a common carrier and a being a publisher. Your right there is no hard legal line but the more they decide what is allowed and what is not allowed the more they move farther away from being a common carrier.

> Social media companies don't become more liable just because they moderate.

Actually they do:

https://cs.stanford.edu/people/eroberts/cs181/projects/1997-....

And also: https://cs.stanford.edu/people/eroberts/cs181/projects/1997-...