I dont agree. The most general word is 'move'. It specifies nothing other than a change of location. By contrast 'pull' is specific in that you must be in front of the object being moved, whilst 'push' means you are behind it. 'Drag' means to pull with high friction, so you could tie a rope around a log and drag it. If you put the log in a cart you would pull the cart (or push it). To 'draw' seems archaic to me, your example phrase is one such place it is still used.
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