Nose wheel shimmy is largely a design fault - the nose wheel leg should slope forward from the vertical by at least 23 degrees. When designed that way a shimmy damper really isn't needed as the wheel tracks properly by itself. Look at how a bicycle's steering is arranged. Riding no hands is easy but try riding one with the handlebars reversed 180 degrees thus making the steering leg effectively vertical or even sloping back.
Knowing the above won't help much in changing an existing design fault but it does partly explain why hauling back on the stick stops shimmying - you are increasing the forward slope nearer to 23 degrees. Another reason pulling the nose up helps is that all three scissor hinges then ride in less worn areas thus taking out some of the play. There are quite a few linkages between the nose-gear leg and the shimmy damper on Cessnas. All these linkages wear, and the sum total of play can be quite large. Some people jack their aircraft, grab the nose-wheel and wriggle it, and conclude that they haven't got much play. However, the nose-wheel is not in this position when it shimmies. It has the weight of the plane on it, and this is why checking for play should not be done while jacked. The best way is to roll the aircraft forward on to two flat sheets of plastic or metal with plenty of grease between the two sheets. This allows you to wriggle the nose wheel leg laterally to determine the amount of play and where it is. It is amazing how much more play there is in the normal running position than when hanging off the jacks. Personally I get my 2 flat sheets by cutting the sides out of a rectangular-shaped 1-litre oil container, which are usually already lubed, and readily available.
Finally, holding the nose high on a Cessna 210 should be mandatory as it helps prevent propellor damage from sucking in rocks. Since I have to pay for my props I would be most unhappy seeing someone startup, taxi, take-off or land my aircraft without the stick being well back.