"Leader" is usually reserved for a different kind of work. A leader determines where the organization should go; a manager ensures that everyone under him is marching in that direction. They require very different skillsets - leading is an outward-facing, synthesizing, strategic role, while managing is an inwards-facing, empathizing, tactical role. To use Ben Horowitz's terminology [1], leaders are Ones and managers are Twos. The CEO's usually the most visible leader in the company, but you'll often find them at lower levels as well, like the tech lead trying out a new experimental Skunkworks project.
Both of them are distinct from "boss" as the grandparent describes it, which is usually what happens when you get a manager who lacks empathy, awareness, and flexibility. Both leader and manager are highly cognitively-complex, pro-active roles that require constant information-gathering. Boss is what happens when you get someone in that position who lacks the confidence or skillset to stay on top of all that information and then reacts through command & control techniques when things go wrong.
"a manager ensures that everyone under him is marching in that direction"
That sounds like a boss to me (and not just because you used an unnecessarily gender-specific word ;). One of the anecdotes in the article is about this person, as a manager, acting as a listening board and ultimately conduit for an engineer when discussing the importance of a specific project at Twitter, and raising questions that came from that engineer that ultimately resulted in the project being cancelled. I think that advocacy role is important management – where the manager advocates both up and down (and ultimately you can't "tell" people what to do, you can only fire them or convince them, so it's "advocacy" both directions).
Difference between a "manager" and "boss" is entirely in how they go about getting everyone on the same page. For any given strategic direction, there are a lot of different ways that could be achieved. A good manager takes in information from all her reports (including their preferences, fears, concerns, and goals) and then figures out a plan that keeps all of them happy while also achieving the strategic goals coming down from above. A bad manager takes the goal as the only input and then outputs commands to achieve it.
Both of them are distinct from "boss" as the grandparent describes it, which is usually what happens when you get a manager who lacks empathy, awareness, and flexibility. Both leader and manager are highly cognitively-complex, pro-active roles that require constant information-gathering. Boss is what happens when you get someone in that position who lacks the confidence or skillset to stay on top of all that information and then reacts through command & control techniques when things go wrong.
[1] http://www.bhorowitz.com/ones_and_twos