Is this the 'Weaver's Knot'?
The knot on the web page you posted looks like a basic square knot (aka reef knot) to me, at least as far as the end result is concerned. (They're tying it upside-down for some reason, but they're ending up with two simple overhand knots, one right, one left, and that's gonna get you a square knot.) One of the square knot's main uses is to join two pieces of thread/yarn/string/rope, so it can certainly be used to tie in a new piece of thread while weaving.
What you're tying looks like maybe a strange configuration of a granny knot, which is not likely to stay put if you pull on the ends.
An actual &quot;weaver's knot&quot; does exist, but doesn't look like either your image or the website you linked to. It's also called a sheet bend, if that helps.