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: 
@AccidentalTaylorExpansion gave the accurate answer and the proper reasoning that the **normal force depends on the intermolecular distances**.

You can visualise it actually in a ground with mud.

When you try to walk, your foot goes into it and after a certain depth you may stop sinking. **This shows that when you came in contact with the surface of mud , your weight was not equal to the normal contact force but when you pressed the mud the atoms came closer and after a certain distance it equalised your weight** and this is the point where you came to rest and ready to take the second step.

Hope it helps .
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.