The Power of Writing
To answer this question, it's important to first ask another:
What is the essence of writing?
In my opinion as a novice writer, writing is a tool by which an individual documents and communicates something—a feeling, an emotion, a fantasy, a world, an idea, an inspiration, something that has the power to change another individual's mindset or their mood. Just as a copper wire conducts physical power from a source to a receiver, writing conducts an emotional power from an artist to someone whom that artist means to influence. The efficacy of a wire is determined by how well it conducts power, how much power it can conserve en route and deliver to its intended recipient. The efficacy of writing can be similarly determined.
Now, to get to the original question: what does it mean to be a good writer? Essentially, a good writer is someone who knows how to properly use writing to conduct their thoughts and feelings, a literary electrician, as it were. The better the writer is able to use these tools, the more their readers are able to peer into the writer's heart, into the writer's mind, and in turn be changed and energized by the ethereal power of a good story or a passionate argument. Writing, just like any physical vessel, is of course an imperfect conductor; however, the aim of any good writer is to continuously strive for better tools, better conductors that can deliver a larger share of that power.
I'd like to take issue with how the second question is posed. I don't subscribe to the school of thought that the two sides are binary or mutually exclusive; rather, good writing does not necessarily stand in opposition to social media. Like any tool, there is a proper way and an improper way to use such tools. Granted, the majority of society does not use modern day tools in a proper literary fashion. But good writers have rarely been in the majority. More often than not, good writers are born from the minority, a rarer breed than your everyday internet troll. Contemporary writers have the powerful opportunity to use social media (Prose. is an additional example of such a tool) in an elevated manner and influence their peers in a way that's unprecedented in human history.