Firstly, acceleration is going on when something is on top of a solid surface. Acceleration due to gravity, however, is equalised by the resistance provided by the solid surface.
There is no acceleration going on, as acceleration is change in speed or direction of travel, and the item in the example was specified to be at rest. Unless you meant to imply that the object would be at rest regarding one point of reference (say, the earth) and accelerating compared to a different frame of reference (such as the sun or the center of the galaxy) in which case that's retarded and serves no purpose besides obfuscation of the point.
Secondly, you are weightless in freefall, as weight is provided by the resistance against gravity.
Even with this new definition (which is totally different from your previous definition) you're wrong. When at terminal velocity, air molecules are exerting resistance against gravitational force equivalent to what would normally be termed the object's weight. That's why there's no acceleration in the first place.
But I tire of this discussion and I suspect you're just trolling anyway, so here are some definitions:
For weight:
Google: A body's relative mass or the quantity of matter contained by it, giving rise to a downward force; the heaviness of a person or thing.
Dictionary.com: the force that gravitation exerts upon a body, equal to the mass of the body times the local acceleration of gravity: commonly taken, in a region of constant gravitational acceleration, as a measure of mass.
TheFreeDictionary.com: The force with which a body is attracted to Earth or another celestial body, equal to the product of the object's mass and the acceleration of gravity.
Wikipedia: In science and engineering, the weight of an object is the force on the object due to gravity.
Merriam-Webster: the force with which a body is attracted toward the earth or a celestial body by gravitation and which is equal to the product of the mass and the local gravitational acceleration
Those make reference to gravitational acceleration, which it now occurs to me you may have been confusing with acceleration due to gravity. Honestly, that would be a fairly understandable mistake. However they are not the same; gravitational acceleration is a pure measure of the force of gravity; it assumes a vacuum. Earth's gravitational acceleration is about 9.8 meters per second per second.