first off: your general problem is that edges do not “know” that at the point of crossing another edge, they need to connect to that point. You have to break them apart to have a “connected” crossing.
to connect the edge with the vertex, you highlighted, do the following:
select the edge, then subdivide it.
the newly created vertex is in the middle of the edge, but you want it to be in the same place as your “unconnected” vertex from the video: select it, turn on snapping to vertices, press G to grab the vertex, then constrain the movement to the axis running along the edge (in your example: press X) then move your mouse over the “unconnected” vertex and confirm.
you now should have two vertices in the same place or very close together: Remove doubles or merge the two (alt+M) to finally get the desired connection between edge and vertex.
however it’s not advisable to construct your mesh that way (i.e. having edges cross each other without vertices connecting them) in the first place, because fixing this for a complex mesh can be very nasty.
hope this helps a bit