Gravity gimbal and effect from overall weight
Gimbals are nothing more than a ball bearing (or a complex series of ball bearings; or even simply a sharp point which rests on a surface which allows it to move on a 2 axis level), which using weights balanced correctly provide a "zero gravity" effect on both the camera and the counterweight. 

This doesn't apply to electronic gimbals which use servos, a microprocessor, and sensors to electronically move the gimbal head. 

That said:

Yes weight will play a role in the overall stability of the camera unit. 

I have taught workshops on camera technique, and I often joke that if you want to get steadier footage from a small lightweight camera (think GoPro), gafftape it to a brick. 

Mass requires energy to move. So the lighter the object, the easier it is to move. Try to shake a marble around with your hand grasping it with two fingers, then do the same with a bowling ball. The motion, if you tracked it, would be both smoother, and more stable - smoother arcs, turns, etc, with the object that has more mass. 

So with a steadicam rig (which I used to own a full blown rig capable of holding cameras in the 30lb range); a heavy camera with a heavy counterweight, which is properly balanced with give you far steadier shots, SPECIFICALLY with vibration. As a large heavy object is hard to "vibrate". So the little shakes, will be minimized by the mass added to the camera. 

What's most important, if using a steadicam type rig is that you have perfect balance, that is, if you turn the camera 90 degrees (thus shooting sideways) the camera will stay in that position. 

You do not want to set it up so the unit is counterweight heavy, keeping the camera upright. 

The camera should be able to "hold" any position, angle, on any axis, and not return to upright. 

If the camera is not balanced this way, you will encounter what is known as steadicam wobble. This is where the balance is off, and by the operators movement the effects of inertia and momentum push and tug differently on the camera vs the counterweight. 

Lastly, the term for non-motorized stabilizers is a camera stabilizer. There is no one term; but steadicam is a goto term used by most. 

Now as far as motorized stabilizers, none of what I said factors in, because the motorized stabilizers I own and have worked with, do not use weights, rather they use servos, and tension systems which differ than their non-electronic counterparts. 

If you are using a light weight camera, and non-electronic steady system; you will achieve better results by adding ballast to both the camera and the counterweight. The ballast you add can be steel plates with holes drilled, cheeseplates, anything that has weight, and that you get the unit balanced to where it is not bottom heavy.