> I don't see how the heck my website is a public square but my home or café isn't, this argument sounds self-contradictory.
Yup. This is the exact problem that we’re (as a society/world) wrestling with.
The reason it is (not just seems) different is because of the scope. A message on a chalkboard cannot reach millions of people (without the internet, ignoring [mass] media because the way it amplified things like this was far more complicated and was intentional), but it can on a website.
That by itself distorts the public/private argument, but we as a society aren’t sure how or to what extent yet.
These lawsuits are the second step (the first step was arguing about it in public) of figuring that out.
(Going back to my side note on media, these arguments will affect media outlets directly/indirectly as well.)