My steering has been whining when turning the wheel left or right for the past year. Felt a lot like what everybody would describe as low steering fluid or a steering pump going bad.
Nothing crazy, but the high pitch sound has been bothering me, and I could not find the reason for it.
Last weekend, I changed the steering damper for a totally unrelated reason, just because I had read here that it improved how the car handled, and it looked like mine had never been done and the car has about 90K miles.
Well, I did not see a big difference in the steering itself, but low and behold, the whining is gone! I did not expect that..

So there, if somebody is experiencing that bothering high pitch whining sound when turning the steering wheel, know that it could be the steering damper, and that changing is indeed very easy and only requires two #16 socket wrenches.

PS: I tried to compare the old and new damper by compressing/depressing them by hand, and could not tell the difference. Wish I had a scale available to compare the compression force.