Not quite, and I'm feeling tired at the moment so I'm struggling to find the right words. It's all about fulcrums and leverage and pi is probably involved at some point.
Let me try and paint a picture. If you take two extremes.
#1 with bars 120cm wide, the circumference of the diameter of the circle that they would turn in (if they could go through 360 degrees) would be around 3.8m, so moving the end of the bars by 2cm would move the front wheel from the straight ahead by less than 1 degree (I think).
#2 with bars 30cm wide, the circumference of the diameter of the circle that they would turn in (if they could go through 360 degrees) would be around 94cm, so moving the end of the bars by 2cm would move the front wheel around 8 degrees from the straight ahead, which is quite a distance.
So, the further you get from the steerer, the larger the diameter of the circle, the less impact any movement at the bars has on the direction of the bike. Thus, addressing your second question first, no.
Moving the hands a given amount will move the front of the bike by less than if you were holding closer to the centre of the circle.
Now, look at it from the steerer rather than the bars. Yes it does mean that the least amount of pressure is required to turn the bars but the other side is that you would need to put a heck of a lot more pressure at the centre of the circle, at the steerer, to be able to move your hands. So, by holding further from the steerer, your bike will be less deflected by movements from the tyre through the bars because you have greater leverage to maintain your line.
Therefore, holding the tops close to the stem makes for a twitchy bike with small inputs creating huge movement in the wheel, whereas holding the hoods or the tops gives you greater leverage, more fine control and a reduced liability to be deflected by steering forces travelling up the forks.
However... all this physics stuff which is making my brain hurt could be irrelevant because (and I always loved this get out clause) it relies on everything else being equal.
The truth is that the two hand positions mean that everything else is not equal. When you are on the tops, there is generally less weight going down your arms and through your hands then if you are on the hoods. I use the top of the levers because that position is even more stretched out than using the hoods but that means that even more weight is taken on my hands and it means that my arms are straighter than when on the tops, so there is less "shock absorption" left at my elbow. It is possible that this difference in weight and the reduction in the ability to absorb bumps do conspire together to mean that a small wobble isn't unknowingly absorbed by your arms but does make the bike twitch.
