Problems with re-stringing

I pull the string tight to the tuning post (so theres no slack in the string from the bridge to the nut), then wind the string three times around the post over lapping the other loops, then carefully guide it through the hole and pull it tight. I use a needle nose plier to pull the excess string (loops)tight to the post. Then I grab the string mid point (by the 12th fret) and pull up taking all the slack out and winding the tuner at the same time.

While I'm pulling up on the string I also work my way up the neck toward the nut stretching the string as it becomes closer to being tuned. I'm not to fond of the way their doing at that site you posted.

Just my way

Joe

I string the string through the bridge and then through the tuner. I kink the string about an inch, maybe a little more, past the tuner, then I pull the string back through until the bend hits the tuner. I then proceed to tune the string near its proper pitch, and I clip off the excess. While I tune it up, I keep enough tension on it, by pulling, that I get good tight winds. I stretch it like Joe does to make it hold its pitch sooner.

I've never tried this knot technique. This seems like it would be difficult with the wound strings. You can simplify this and get increased tuning stability:
Do what they have done in the first 2 pictures, so that the loose end has been brought back under the string that will wrap around the post.
Then, hold the string up as you wind so that the loose end gets trapped under the first winding, just where it enters the post. This acts as your knot since the string tension traps this end.
Wind the string as always, spiraling down the post, the trim to length.
That weblink implies that that other technique is for nylon strings. I've had great luck with the technique I describe and I can be a bit whammy-happy.
As Forrok says, you will probably need to make a couple initial winding around the post (below the holes) in order to get the 2 or 3 wraps recommended for tuning stability. They don't show these in either picture.

I use this technique when restringing.

http://www.frets.com/FRETSPages/Musician/Guitar/Setup/SteelStrings/Stringing/ststringing1.html

I've used the method shown in Steve-o's link and it works fine, there's absolutely no problem with tuning, even drop tuning works.
There is one caveat - take great care with kinking strings. If you do it too hard you will weaken the string and cause it to break when put under tension. This has only happened once to me and it was a .010".