Question: What do we mean by the term [weight](https://en.wikipedia.org/wiki/Weight)? Is it an upward force exerted by an object to counteract the downward force of gravity (or) is it the downward force itself? Is the magnitude of the weight always equal to the gravitational force regardless of the frame?

Answer: 
It is a consequence of the 3rd law. The man pushes down on the floor, with his weight, and the floor pushes back with an equal and opposite normal force.
When you stand still on the floor, are you accelerating toward the center of the Earth?

If your acceleration is zero, that must mean that the net forces acting on your body are zero. That must mean, that the contact force pushing up against your feet must be equal and opposite to the force of gravity pulling you down.