No, "most people" do not consider that to be what socialism is. Particularly those of us who live in countries with the aforementioned policies. Here we've had real socialists who wanted to take away our fundamental individual rights, amongst them the right to ownership, which frankly is a scary idea.

A lot of our regulations and limits on the free market don't have a socialist bent at all, but are intended to defend our individual liberties against large corporations, which if left unchecked can become corporate institutions, something the US has fallen victim to.

I'd consider these policies as important, if not moreso than our social welfare systems. The social mobility and safety provided by these are meaningless if an arbitrary decision by google, amazon or some bank can singlehandedly ruin your life.