Yes, the earth does precess about its axis:
http://www-istp.gsfc.nasa.gov/stargaze/Sprecess.htm
And as you can read on that site some see a link between the precession and ice ages. But, the earths north magnetic pole has moved nearly 1,100 km in the last century:
http://www.physorg.com/news8917.html
I am no expert on climate, but I would suspect if the true axis of rotation of the earth had moved that far there would be huge climatic changes.
In case the sites you posted didn't mention it, it's important to clarify that the true north pole (axis of rotation) isn't what's moving during precession (that is known as true polar wander and is very slight [15 miles over a million years?]), and this topic is totally unrelated to the magnetic-pole wandering. The rotation is still around the same axis--the same north and south points on the globe.
But, as you wrote, the earth is wobbling in space such that the axis was pointing generally toward Vega as the glaciers were receding 12,000 years ago and by ancient Egyptian times, it was pointing toward Thuban. It's near Polaris now, and will be pointing toward Vega again as our north star about 14,000 years in the future. Of course, after that, the circuit will take it back to Thuban. (Contrary to the claims of some misunderstanding New Agers and Catastrophists, scientists aren't claiming it will "jump" or "shift" to these new stars--they are just markers on the circuit it takes in its wobble.)
So the precession of the axis of rotation itself in space is much slower than the movement of the magnetic pole on the earth itself, relative to the axis of rotation. If the axis of rotation had moved 20 degrees down to where the magnetic pole was early last century, we'd have larger problems than a little debate on climate change!
*Note that besides precession, there are other orbital cycles that add to the climate forcings that must be taken into account (eccentricity, obliquity, orbit inclination, elliptical precession, maybe others?). I'm likewise not an orbital mechanics or climatology expert, but in a former life I was doing climatology, so I think I got this mostly right from memory. Still, confirmation is up to the reader, as I'm the very model of a modern Internet commentator--totally lost without web access!